Polynomials and Factoring; More on Probability


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1 Polynomials and Factoring; More on Probability Melissa Kramer, (MelissaK) Anne Gloag, (AnneG) Andrew Gloag, (AndrewG) Say Thanks to the Authors Click (No sign in required)
2 To access a customizable version of this book, as well as other interactive content, visit AUTHORS Melissa Kramer, (MelissaK) Anne Gloag, (AnneG) Andrew Gloag, (AndrewG) CK12 Foundation is a nonprofit organization with a mission to reduce the cost of textbook materials for the K12 market both in the U.S. and worldwide. Using an opencontent, webbased collaborative model termed the FlexBook, CK12 intends to pioneer the generation and distribution of highquality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform. Copyright 2013 CK12 Foundation, The names CK12 and CK12 and associated logos and the terms FlexBook and FlexBook Platform (collectively CK12 Marks ) are trademarks and service marks of CK12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK12 Content (including CK12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/Non Commercial/Share Alike 3.0 Unported (CC BYNCSA) License ( as amended and updated by Creative Commons from time to time (the CC License ), which is incorporated herein by this reference. Complete terms can be found at Printed: August 21, 2013 EDITOR Annamaria (AnnamariaF) Farbizio,
3 Chapter 1. Polynomials and Factoring; More on Probability C HAPTER 1 Polynomials and Factoring; More on Probability C HAPTER O UTLINE 1.1 Addition and Subtraction of Polynomials 1.2 Multiplication of Polynomials 1.3 Special Products of Polynomials 1.4 Polynomial Equations in Factored Form 1.5 Factoring Quadratic Expressions 1.6 Factoring Special Products 1.7 Factoring Polynomials Completely 1.8 Probability of Compound Events 1.9 Chapter 9 Review 1.10 Chapter 9 Test This chapter will present a new type of function: the polynomial. Chances are, polynomials will be new to you. However, polynomials are used in many careers and real life situations  to model the population of a city over a century, to predict the price of gasoline, and to predict the volume of a solid. This chapter will also present basic factoring  breaking a polynomial into its linear factors. This will help you solve many quadratic equations found in Chapter 10. 1
4 1.1. Addition and Subtraction of Polynomials Addition and Subtraction of Polynomials So far we have discussed linear functions and exponential functions. This lesson introduces polynomial functions. Definition: A polynomial is an expression made with constants, variables, and positive integer exponents of the variables. An example of a polynomial is: 4x 3 + 2x 2 3x + 1. There are four terms: 4x 3, 2x 2, 3x, and 1. The numbers appearing in each term in front of the variable are called the coefficients. 4, 2, and 3 are coefficients because those numbers are in front of a variable. The number appearing all by itself without a variable is called a constant. 1 is the constant because it is by itself. Example 1: Identify the following expressions as polynomials or nonpolynomials. (a) 5x 2 2x (b) 3x 2 2x 2 (c) x x 1 (d) 5 x 3 +1 (e) 4x 1 3 (f) 4xy 2 2x 2 y 3 + y 3 3x 3 Solution: (a) 5x 2 2x This is a polynomial. (b) 3x 2 2x 2 This is not a polynomial because it has a negative exponent. (c) x x 1 This is not a polynomial because is has a square root. 5 (d) This is not a polynomial because the power of x appears in the denominator. x 3 +1 (e) 4x 1 3 This is not a polynomial because it has a fractional exponent. (f) 4xy 2 2x y 3 + y 3 3x 3 This is a polynomial. Classifying Polynomials by Degree The degree of a polynomial is the largest exponent of a single term. 4x 3 has a degree of 3 and is called a cubic term or 3 rd order term. 2x 2 has a degree of 2 and is called a quadratic term or 2 nd order term. 3x has a degree of 1 and is called a linear term or 1 st order term. 1 has a degree of 0 because there is no variable. Polynomials can have more than one variable. Here is another example of a polynomial: t 4 6s 3 t 2 12st + 4s 4 5. This is a polynomial because all exponents on the variables are positive integers. This polynomial has five terms. Note: The degree of a term is the sum of the powers on each variable in the term. t 4 has a degree of 4, so it s a 4 th order term. 6s 3 t 2 has a degree of 5, so it s a 5 th order term. 2
5 Chapter 1. Polynomials and Factoring; More on Probability 12 st has a degree of 2, so it s a 2 nd order term. 4s 4 has a degree of 4, so it s a 4 th order term. 5 is a constant, so its degree is 0. Since the highest degree of a term in this polynomial is 5, this is a polynomial of degree 5 or a 5 th order polynomial. Example 2: Identify the coefficient on each term, the degree of each term, and the degree of the polynomial. x 4 3x 3 y 2 + 8x 12 Solution: The coefficients of each term in order are 1, 3, 8 and the constant is 12. The degrees of each term are 4, 5, 1, and 0. Therefore, the degree of the polynomial is 5. A monomial is a onetermed polynomial. It can be a constant, a variable, or a combination of constants and variables. Examples of monomials are: b 2 ; 6; 2ab 2 ; 1 4 x2 Rewriting Polynomials in Standard Form Often, we arrange the terms in a polynomial in standard from in which the term with the highest degree is first and is followed by the other terms in order of decreasing power. The first term of a polynomial in this form is called the leading term, and the coefficient in this term is called the leading coefficient. Example 3: Rearrange the terms in the following polynomials so that they are in standard form. Indicate the leading term and leading coefficient of each polynomial. (a) 7 3x 3 + 4x (b) ab a 3 + 2b Solution: (a) 7 3x 3 + 4x is rearranged as 3x 3 + 4x + 7. The leading term is 3x 3 and the leading coefficient is 3. (b) ab a 3 + 2b is rearranged as a 3 + ab + 2b. The leading term is a 3 and the leading coefficient is 1. Simplifying Polynomials A polynomial is simplified if it has no terms that are alike. Like terms are terms in the polynomial that have the same variable(s) with the same exponents, but they can have different coefficients. 2x 2 y and 5x 2 y are like terms. 6x 2 y and 6xy 2 are not like terms. If we have a polynomial that has like terms, we simplify by combining them. x 2 + 6xy 4xy + y 2 Like terms 3
6 1.1. Addition and Subtraction of Polynomials This polynomial is simplified by combining the like terms 6xy 4xy = 2xy. We write the simplified polynomial as x 2 + 2xy + y 2. Example 4: Simplify by collecting and combining like terms. a 3 b 3 5ab 4 + 2a 3 b a 3 b 3 + 3ab 4 a 2 b Solution: Use the Commutative Property of Addition to reorganize like terms then simplify. = (a 3 b 3 a 3 b 3 ) + ( 5ab 4 + 3ab 4 ) + 2a 3 b a 2 b = 0 2ab 4 + 2a 3 b a 2 b = 2ab 4 + 2a 3 b a 2 b Adding and Subtracting Polynomials To add or subtract polynomials, you have to group the like terms together and combine them to simplify. Example 5: Add and simplify 3x 2 4x + 7 and 2x 3 4x 2 6x + 5. Solution: Add 3x 2 4x + 7 and 2x 3 4x 2 6x + 5. (3x 2 4x + 7) + (2x 3 4x 2 6x + 5) = 2x 3 + (3x 2 4x 2 ) + ( 4x 6x) + (7 + 5) = 2x 3 x 2 10x + 12 Multimedia Link: For more explanation of polynomials, visit  Purplemath s website. Example 6: Subtract 5b 2 2a 2 from 4a 2 8ab 9b 2. Solution: (4a 2 8ab 9b 2 ) (5b 2 2a 2 ) = [(4a 2 ( 2a 2 )] + ( 9b 2 5b 2 ) 8ab = 6a 2 14b 2 8ab Solving RealWorld Problems Using Addition or Subtraction of Polynomials Polynomials are useful for finding the areas of geometric objects. In the following examples, you will see this usefulness in action. Example 7: Write a polynomial that represents the area of each figure shown. (a) 4
7 Chapter 1. Polynomials and Factoring; More on Probability (b) Solution: The blue square has area: y y = y 2. The yellow square has area: x x = x 2. The pink rectangles each have area: x y = xy. Test area = y 2 + x 2 + xy + xy = y 2 + x 2 + 2xy To find the area of the green region we find the area of the big square and subtract the area of the little square. The big square has area y y = y 2. The little square has area x x = x 2. Area of the green region = y 2 x 2 Practice Set Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra:Addition and Subtraction ofpolynomials (15:59) MEDIA Click image to the left for more content. 5
8 1.1. Addition and Subtraction of Polynomials Define the following key terms. 1. Polynomial 2. Monomial 3. Degree 4. Leading coefficient For each of the following expressions, decide whether it is a polynomial. Explain your answer. 5. x 2 + 3x x2 y 9y x t2 1 t 2 Express each polynomial in standard form. Give the degree of each polynomial x 10. 8x 4 x + 5x x x + 3x f 8 7 f x 5x 2 + 8x x 2 9x Add and simplify. 15. (x + 8) + ( 3x 5) 16. (8r 4 6r 2 3r + 9) + (3r 3 + 5r r 9) 17. ( 2x 2 + 4x 12) + (7x + x 2 ) 18. (2a 2 b 2a + 9) + (5a 2 b 4b + 5) 19. (6.9a 2 2.3b 2 + 2ab) + (3.1a 2.5b 2 + b) Subtract and simplify. 20. ( t + 15t 2 ) (5t 2 + 2t 9) 21. ( y 2 + 4y 5) (5y 2 + 2y + 7) 22. ( h 7 + 2h h 3 + 4h 2 h 1) ( 3h h 3 3h 2 + 8h 4) 23. ( 5m 2 m) (3m 2 + 4m 5) 24. (2a 2 b 3ab 2 + 5a 2 b 2 ) (2a 2 b 2 + 4a 2 b 5b 2 ) Find the area of the following figures
9 Chapter 1. Polynomials and Factoring; More on Probability Mixed Review 29. Solve by graphing { y = 1 3 x 4 y = 4x Solve for u: 12 = 4 u. 31. Graph y = x 4 +3 on a coordinate plane. a. State its domain and range. b. How has this graph been shifted from the parent function f (x) = x? 32. Two dice are rolled. The sum of the values are recorded. a. Define the sample space. b. What is the probability the sum of the dice is nine? 33. Consider the equation y = 6500(0.8) x. a. Sketch the graph of this function. b. Is this exponential growth or decay? c. What is the initial value? d. What is its domain and range? e. What is the value when x = 9.5? 34. Write an equation for the line that is perpendicular to y = 5 and contains the ordered pair (6, 5) 7
10 1.2. Multiplication of Polynomials Multiplication of Polynomials When multiplying polynomials together, we must remember the exponent rules we learned in the last chapter, such as the Product Rule. This rule says that if we multiply expressions that have the same base, we just add the exponents and keep the base unchanged. If the expressions we are multiplying have coefficients and more than one variable, we multiply the coefficients just as we would any number. We also apply the product rule on each variable separately. Example: (2x 2 y 3 ) (3x 2 y) = (2 3) (x 2 x 2 ) (y 3 y) = 6x 4 y 4 Multiplying a Polynomial by a Monomial This is the simplest of polynomial multiplications. Problems are like that of the one above. Example 1: Multiply the following monomials. (a) (2x 2 )(5x 3 ) (c) (3xy 5 )( 6x 4 y 2 ) (d) ( 12a 2 b 3 c 4 )( 3a 2 b 2 ) Solution: (a) (2x 2 )(5x 3 ) = (2 5) (x 2 x 3 ) = 10x 2+3 = 10x 5 (c) (3xy 5 )( 6x 4 y 2 ) = 18x 1+4 y 5+2 = 18x 5 y 7 (d) ( 12a 2 b 3 c 4 )( 3a 2 b 2 ) = 36a 2+2 b 3+2 c 4 = 36a 4 b 5 c 4 To multiply monomials, we use the Distributive Property. Distributive Property: For any expressions a, b, and c, a(b + c) = ab + ac. This property can be used for numbers as well as variables. This property is best illustrated by an area problem. We can find the area of the big rectangle in two ways. One way is to use the formula for the area of a rectangle. Area o f the big rectangle = Length Width Length = a, Width = b + c Area = a (b + c) 8
11 Chapter 1. Polynomials and Factoring; More on Probability The area of the big rectangle can also be found by adding the areas of the two smaller rectangles. This means that a(b + c) = ab + ac. Area o f red rectangle = ab Area o f blue rectangle = ac Area o f big rectangle = ab + ac In general, if we have a number or variable in front of a parenthesis, this means that each term in the parenthesis is multiplied by the expression in front of the parenthesis. a(b + c + d + e + f +...) = ab + ac + ad + ae + a f +... The... means and so on. Example 2: Multiply 2x 3 y( 3x 4 y 2 + 2x 3 y 10x 2 + 7x + 9). Solution: 2x 3 y( 3x 4 y 2 + 2x 3 y 10x 2 + 7x + 9) = (2x 3 y)( 3x 4 y 2 ) + (2x 3 y)(2x 3 y) + (2x 3 y)( 10x 2 ) + (2x 3 y)(7x) + (2x 3 y)(9) = 6x 7 y 3 + 4x 6 y 2 20x 5 y + 14x 4 y + 18x 3 y Multiplying a Polynomial by a Binomial A binomial is a polynomial with two terms. The Distributive Property also applies for multiplying binomials. Let s think of the first parentheses as one term. The Distributive Property says that the term in front of the parentheses multiplies with each term inside the parentheses separately. Then, we add the results of the products. (a + b)(c + d) = (a + b) c + (a + b) d Let s rewrite this answer as c (a + b) + d (a + b) We see that we can apply the Distributive Property on each of the parentheses in turn. c (a + b) + d (a + b) = c a + c b + d a + d b (or ca + cb + da + db) What you should notice is that when multiplying any two polynomials, every term in one polynomial is multiplied by every term in the other polynomial. Example: Multiply and simplify (2x + 1)(x + 3). Solution: We must multiply each term in the first polynomial with each term in the second polynomial. First, multiply the first term in the first parentheses by all the terms in the second parentheses. Now we multiply the second term in the first parentheses by all terms in the second parentheses and add them to the previous terms. 9
12 1.2. Multiplication of Polynomials Now we can simplify. (2x)(x) + (2x)(3) + (1)(x) + (1)(3) = 2x 2 + 6x + x + 3 = 2x 2 + 7x + 3 Multimedia Link: For further help, visit Purplemath s website or watch this CK12 Basic Algebra:Adding and Subtracting Polynomials MEDIA Click image to the left for more content. YouTube video. Example 3: Multiply and simplify (4x 5)(x 20). Solution: (4x)(x) + (4x)( 20) + ( 5)(x) + ( 5)( 20) = 4x 2 80x 5x = 4x 2 85x Solving RealWorld Problems Using Multiplication of Polynomials We can use multiplication to find the area and volume of geometric shapes. Look at these examples. Example 4: Find the area of the following figure. Solution: We use the formula for the area of a rectangle: Area = length width. For the big rectangle: Length = B + 3, Width = B + 2 Area = (B + 3)(B + 2) = B 2 + 2B + 3B + 6 = B 2 + 5B + 6 Example 5: Find the volume of the following figure. 10
13 Chapter 1. Polynomials and Factoring; More on Probability Solution: T he volume o f this shape = (area o f the base) (height). Area of the base = x(x + 2) = x 2 + 2x Volume = (area o f base) height Volume = (x 2 + 2x)(2x + 1) You are asked to finish this example in the practice questions. Practice Set Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra:Multiplication of Polynomials (9:49) MEDIA Click image to the left for more content. Multiply the following monomials. 1. (2x)( 7x) 2. 4( 6a) 3. ( 5a 2 b)( 12a 3 b 3 ) 4. ( 5x)(5y) 5. y(xy 4 ) 6. (3xy 2 z 2 )(15x 2 yz 3 ) Multiply and simplify. 11
14 1.2. Multiplication of Polynomials 7. x 8 (xy 3 + 3x) 8. 2x(4x 5) 9. 6ab( 10a 2 b 3 + c 5 ) 10. 9x 3 (3x 2 2x + 7) 11. 3a 2 b(9a 2 4b 2 ) 12. (x 2)(x + 3) 13. (a + 2)(2a)(a 3) 14. ( 4xy)(2x 4 yz 3 y 4 z 9 ) 15. (x 3)(x + 2) 16. (a 2 + 2)(3a 2 4) 17. (7x 2)(9x 5) 18. (2x 1)(2x 2 x + 3) 19. (3x + 2)(9x 2 6x + 4) 20. (a 2 + 2a 3)(a 2 3a + 4) 21. (3m + 1)(m 4)(m + 5) 22. Finish the volume example from Example 5 of the lesson. Volume = (x 2 + 2x)(2x + 1) Find the areas of the following figures Find the volumes of the following figures
15 Chapter 1. Polynomials and Factoring; More on Probability Mixed Review 27. Give an example of a fourth degree trinomial in the variable n. 28. Find the next four terms of the sequence 1, 3 2, 9 4, 28 8, Reece reads three books per week. a. Make a table of values for weeks zero through six. b. Fit a model to this data. c. When will Reece have read 63 books? 30. Write 0.062% as a decimal. 31. Evaluate ab ( a + b 4) when a = 4 and b = Solve for s: 3s(3 + 6s) + 6(5 + 3s) = 21s. 13
16 1.3. Special Products of Polynomials Special Products of Polynomials When we multiply two linear (degree of 1) binomials, we create a quadratic (degree of 2) polynomial with four terms. The middle terms are like terms so we can combine them and simplify to get a quadratic or 2 nd degree trinomial (polynomial with three terms). In this lesson, we will talk about some special products of binomials. Finding the Square of a Binomial A special binomial product is the square of a binomial. Consider the following multiplication: (x + 4)(x + 4). We are multiplying the same expression by itself, which means that we are squaring the expression. This means that: (x + 4)(x + 4) = (x + 4) 2 (x + 4)(x + 4) = x 2 + 4x + 4x + 16 = x 2 + 8x + 16 This follows the general pattern of the following rule. Square of a Binomial: (a + b) 2 = a 2 + 2ab + b 2, and (a b) 2 = a 2 2ab + b 2 Stay aware of the common mistake (a + b) 2 = a 2 + b 2. To see why (a + b) 2 a 2 + b 2, try substituting numbers for a and b into the equation (for example, a = 4 and b = 3), and you will see that it is not a true statement. The middle term, 2ab, is needed to make the equation work. Example 1: Simplify by multiplying: (x + 10) 2. Solution: Use the square of a binomial formula, substituting a = x and b = 10 (a + b) 2 = a 2 + 2ab + b 2 (x + 10) 2 = (x) 2 + 2(x)(10) + (10) 2 = x x Finding the Product of Binomials Using Sum and Difference Patterns Another special binomial product is the product of a sum and a difference of terms. For example, let s multiply the following binomials. (x + 4)(x 4) = x 2 4x + 4x 16 = x 2 16 Notice that the middle terms are opposites of each other, so they cancel out when we collect like terms. This always happens when we multiply a sum and difference of the same terms. 14
17 Chapter 1. Polynomials and Factoring; More on Probability (a + b)(a b) = a 2 ab + ab b 2 = a 2 b 2 When multiplying a sum and difference of the same two terms, the middle terms cancel out. We get the square of the first term minus the square of the second term. You should remember this formula. Sum and Difference Formula: (a + b)(a b) = a 2 b 2 Example 2: Multiply the following binomias and simplify. (5x + 9)(5x 9) Solution: Use the above formula, substituting a = 5x and b = 9. Multiply. (5x + 9)(5x 9) = (5x) 2 (9) 2 = 25x 2 81 Solving RealWorld Problems Using Special Products of Polynomials Let s now see how special products of polynomials apply to geometry problems and to mental arithmetic. Look at the following example. Example: Find the area of the square. Solution: T he area o f the square = side side Area = (a + b)(a + b) = a 2 + 2ab + b 2 Notice that this gives a visual explanation of the square of binomials product. Area o f big square : (a + b) 2 = Area o f blue square = a (area o f yellow) = 2ab + area o f red square = b 2 The next example shows how to use the special products in doing fast mental calculations. Example 3: Find the products of the following numbers without using a calculator. (a)
18 1.3. Special Products of Polynomials (b) 45 2 Solution: The key to these mental tricks is to rewrite each number as a sum or difference of numbers you know how to square easily. (a) Rewrite 43 = (50 7) and 57 = (50 + 7). Then = (50 7)(50 + 7) = (50) 2 (7) 2 = = 2,451. (b) 45 2 = (40 + 5) 2 = (40) 2 + 2(40)(5) + (5) 2 = = 2,025 Practice Set Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra:SpecialProducts ofbinomials (10:36) MEDIA Click image to the left for more content. Use the special product for squaring binomials to multiply these expressions. 1. (x + 9) 2 2. (x 1) 2 3. (2y + 6) 2 4. (3x 7) 2 5. (7c + 8) 2 6. (9a 2 + 6) 2 7. (b 2 1) 2 8. (m 3 + 4) ( t + 2) (6k 3) (a 3 7) (4x 2 + y 2 ) (8x 3) 2 Use the special product of a sum and difference to multiply these expressions (2x 1)(2x + 1) 15. (2x 3)(2x + 3) 16. (4 + 6x)(4 6x) 17. (6 + 2r)(6 2r) 18. ( 2t + 7)(2t + 7) 19. (8z 8)(8z + 8) 20. (3x 2 + 2)(3x 2 2)
19 Chapter 1. Polynomials and Factoring; More on Probability 21. (x 12)(x + 12) 22. (5a 2b)(5a + 2b) 23. (ab 1)(ab + 1) Find the area of the orange square in the following figure. It is the lower right shaded box. 24. Multiply the following numbers using the special products Mixed Review 32. Simplify 5x(3x + 5) + 11( 7 x). 33. Cal High School has grades nine through twelve. Of the school s student population, 1 4 are freshmen, 2 5 are sophomores, 1 6 are juniors, and 130 are seniors. To the nearest whole person, how many students are in the sophomore class? 34. Kerrie is working at a toy store and must organize 12 bears on a shelf. In how many ways can this be done? 35. Find the slope between ( 3 4,1) and ( 3 4, 16). 36. If 1 lb = 454 grams, how many kilograms does a 260pound person weigh? 37. Solve for v: 16 v = Is y = x 4 + 3x a function? Use the definition of a function to explain. 17
20 1.4. Polynomial Equations in Factored Form Polynomial Equations in Factored Form We have been multiplying polynomials by using the Distributive Property, where all the terms in one polynomial must be multiplied by all terms in the other polynomial. In this lesson, you will start learning how to do this process using a different method called factoring. Factoring: Take the factors that are common to all the terms in a polynomial. Then multiply the common factors by a parenthetical expression containing all the terms that are left over when you divide out the common factors. Let s look at the areas of the rectangles again: Area = length width. The total area of the figure on the right can be found in two ways. Method 1: Find the areas of all the small rectangles and add them. Blue rectangle = ab Orange rectangle = ac Red rectangle = ad Green rectangle = ae Purple rectangle = 2a Total area = ab + ac + ad + ae + 2a Method 2: Find the area of the big rectangle all at once. Length = a Width = b + c + d + e + 2 Area = a(b + c + d + e = 2) The answers are the same no matter which method you use: ab + ac + ad + ae + 2a = a(b + c + d + e + 2) 18
21 Chapter 1. Polynomials and Factoring; More on Probability Using the Zero Product Property Polynomials can be written in expanded form or in factored form. Expanded form means that you have sums and differences of different terms: 6x 4 + 7x 3 26x 2 17x + 30 Notice that the degree of the polynomial is four. The factored form of a polynomial means it is written as a product of its factors. The factors are also polynomials, usually of lower degree. Here is the same polynomial in factored form. (x 1)(x + 2)(2x 3)(3x + 5) Suppose we want to know where the polynomial 6x 4 +7x 3 26x 2 17x+30 equals zero. It is quite difficult to solve this using the methods we already know. However, we can use the Zero Product Property to help. Zero Product Property: The only way a product is zero is if one or both of the terms are zero. By setting the factored form of the polynomial equal to zero and using this property, we can easily solve the original polynomial. (x 1)(x + 2)(2x 3)(3x + 5) = 0 According to the property, for the original polynomial to equal zero, we have to set each term equal to zero and solve. (x 1) = 0 x = 1 (x + 2) = 0 x = 2 (2x 3) = 0 x = 3 2 (3x + 5) = 0 x = 5 3 The solutions to 6x 4 + 7x 3 26x 2 17x + 30 = 0 are x = 2, 5 3,1, 3 2. Multimedia Link: For further explanation of the Zero Product Property, watch this CK12 Basic Algebra:Zero Pr oduct Property MEDIA Click image to the left for more content. 19
22 1.4. Polynomial Equations in Factored Form  YouTube video. Example 2: Solve (x 9)(3x + 4) = 0. Solution: Separate the factors using the Zero Product Property: (x 9)(3x + 4) = 0. x 9 = 0 or 3x + 4 = 0 x = 9 3x = 4 x = 4 3 Finding the Greatest Common Monomial Factor Once we get a polynomial in factored form, it is easier to solve the polynomial equation. But first, we need to learn how to factor. Factoring can take several steps because we want to factor completely so we cannot factor any more. A common factor can be a number, a variable, or a combination of numbers and variables that appear in every term of the polynomial. When a common factor is factored from a polynomial, you divide each term by the common factor. What is left over remains in parentheses. Example 3: Factor: 1. 15x a + 9b + 6 Solution: 1. We see that the factor of 5 divides evenly from all terms. 15x 25 = 5(3x 5) 2. We see that the factor of 3 divides evenly from all terms. 3a + 9b + 6 = 3(a + 3b + 2) Now we will use examples where different powers can be factored and there is more than one common factor. Example 4: Find the greatest common factor. (a) a 3 3a 2 + 4a (b) 5x 3 y 15x 2 y xy 3 Solution: (a) Notice that the factor a appears in all terms of a 3 3a 2 + 4a but each term has a different power of a. The common factor is the lowest power that appears in the expression. In this case the factor is a. Let s rewrite a 3 3a 2 + 4a = a(a 2 ) + a( 3a) + a(4) Factor a to get a(a 2 3a + 4) (b) The common factors are 5xy. When we factor 5xy, we obtain 5xy(x 2 3xy + 5y 2 ). 20
23 Chapter 1. Polynomials and Factoring; More on Probability Solving Simple Polynomial Equations by Factoring We already saw how we can use the Zero Product Property to solve polynomials in factored form. Here you will learn how to solve polynomials in expanded form. These are the steps for this process. Step 1: Rewrite the equation in standard form such that: Polynomial expression = 0. Step 2: Factor the polynomial completely. Step 3: Use the zeroproduct rule to set each factor equal to zero. Step 4: Solve each equation from step 3. Step 5: Check your answers by substituting your solutions into the original equation. Example 5: Solve the following polynomial equation. x 2 2x = 0 Solution: x 2 2x = 0 Rewrite: This is not necessary since the equation is in the correct form. Factor: The common factor is x, so this factors as: x(x 2) = 0. Set each factor equal to zero. x = 0 or x 2 = 0 Solve: x = 0 or x = 2 Check: Substitute each solution back into the original equation. Answer x = 0, x = 2 x = 0 (0) 2 2(0) = 0 x = 2 (2) 2 2(2) = 0 Practice Set Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra:Polynomial Equations infactored Form (9:29) 21
24 1.4. Polynomial Equations in Factored Form MEDIA Click image to the left for more content. 1. What is the Zero Product Property? How does this simplify solving complex polynomials? Factor the common factor from the following polynomials a 2 + 9a 3 6a 7 3. yx 3 y x + 16y 4. 3x 3 21x 5. 5x x x x 2 2x 7. 10x x 5 4x xy + 24xy xy a 3 7a y y xy 2 z + 4x 3 y Why can t the Zero Product Property be used to solve the following polynomials? 12. (x 2)(x) = (x + 6) + (3x 1) = (x 3 )(x + 7) = (x + 9) (6x 1) = (x 4 )(x 2 1) = 0 Solve the following polynomial equations. 17. x(x + 12) = (2x + 3)(5x 4) = (2x + 1)(2x 1) = x 2 4x = m = 45m (x 5)(2x + 7)(3x 4) = x(x + 9)(7x 20) = y 3y 2 = x 2 = 27x 26. 4a 2 + a = b 2 5 3b = 0 Mixed Review Rewrite in standard form: 4x + 11x 4 6x x 2. State the polynomial s degree and leading coefficient. 29. Simplify (9a 2 8a + 11a 3 ) (3a a 5 12a) + (9 3a 5 13a). 30. Multiply 1 3 a3 by (36a 4 + 6). 31. Melissa made a trail mix by combining x ounces of a 40% cashew mixture with y ounces of a 30% cashew mixture. The result is 12 ounces of cashews.
25 Chapter 1. Polynomials and Factoring; More on Probability a. Write the equation to represent this situation. b. Graph using its intercepts. c. Give three possible combinations to make this sentence true. 32. Explain how to use mental math to simplify 8(12.99). 23
26 1.5. Factoring Quadratic Expressions Factoring Quadratic Expressions In this lesson, we will learn how to factor quadratic polynomials for different values of a, b, and c. In the last lesson, we factored common monomials, so you already know how to factor quadratic polynomials where c = 0. Factoring Quadratic Expressions in Standard From Quadratic polynomials are polynomials of degree 2. The standard form of a quadratic polynomial is ax 2 + bx + c, where a, b, and c are real numbers. Example 1: Factor x 2 + 5x + 6. Solution: We are looking for an answer that is a product of two binomials in parentheses: (x + )(x + ). To fill in the blanks, we want two numbers m and n that multiply to 6 and add to 5. A good strategy is to list the possible ways we can multiply two numbers to give us 6 and then see which of these pairs of numbers add to 5. The number six can be written as the product of. 6 = 1 6 and = 7 6 = 2 3 and = 5 So the answer is (x + 2)(x + 3). We can check to see if this is correct by multiplying (x + 2)(x + 3). x is multiplied by x and 3 = x 2 + 3x. 2 is multiplied by x and 3 = 2x + 6. Combine the like terms: x 2 + 5x + 6. Example 2: Factor x 2 6x + 8. Solution: We are looking for an answer that is a product of the two parentheses (x + )(x + ). The number 8 can be written as the product of the following numbers. 8 = 1 8 and = 9 Notice that these are two different choices. 8 = ( 1)( 8) and 1 + ( 8) = 9 8 = 2 4 and = 6 And 8 = ( 2) ( 4) and 2 + ( 4) = 6 This is the correct choice. The answer is (x 2)(x 4). Example 3: Factor x 2 + 2x 15. Solution: We are looking for an answer that is a product of two parentheses (x ± )(x ± ). 24
27 Chapter 1. Polynomials and Factoring; More on Probability In this case, we must take the negative sign into account. The number 15 can be written as the product of the following numbers. 15 = 1 15 and = 14 Notice that these are two different choices. And also, 15 = 1 ( 15) and 1 + ( 15) = 14 Notice that these are two different choices. 15 = ( 3) 5 and ( 3) + 5 = 2 This is the correct choice. 15 = 3 ( 5) and 3 + ( 5) = 2 The answer is (x 3)(x + 5). Example 4: Factor x 2 + x + 6. Solution: First factor the common factor of 1 from each term in the trinomial. Factoring 1 changes the signs of each term in the expression. x 2 + x + 6 = (x 2 x 6) We are looking for an answer that is a product of two parentheses (x ± )(x ± ). Now our job is to factor x 2 x 6. The number 6 can be written as the product of the following numbers. 6 = ( 1) 6 and ( 1) + 6 = 5 6 = 1 ( 6) and 1 + ( 6) = 5 6 = ( 2) 3 and ( 2) + 3 = 1 6 = 2 ( 3) and 2 + ( 3) = 1 T his is the correct choice. The answer is (x 3)(x + 2). To Summarize: A quadratic of the form x 2 + bx + c factors as a product of two parenthesis (x + m)(x + n). If b and c are positive then both m and n are positive. Example x 2 + 8x + 12 factors as (x + 6)(x + 2). If b is negative and c is positive then both m and n are negative. Example x 2 6x + 8 factors as (x 2)(x 4). If c is negative then either m is positive and n is negative or viceversa. Example x 2 + 2x 15 factors as (x + 5)(x 3). Example x x 35 factors as (x + 35)(x 1). If a = 1, factor a common factor of 1 from each term in the trinomial and then factor as usual. The answer will have the form (x + m)(x + n). Example x 2 + x + 6 factors as (x 3)(x + 2). 25
28 1.5. Factoring Quadratic Expressions Practice Set Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra:Factoring QuadraticEquations (16:30) MEDIA Click image to the left for more content. Factor the following quadratic polynomials. 1. x x x x x x x x x 2 11x x 2 13x x 2 14x x 2 9x x 2 + 5x x 2 + 6x x 2 + 7x x 2 + 4x x 2 12x x 2 5x x 2 3x x 2 x x 2 2x x 2 5x x x x x x x x x x x x 2 17x 60 Mixed Review Evaluate f (2) when f (x) = 1 2 x2 6x The Nebraska Department of Roads collected the following data regarding mobile phone distractions in traffic crashes by teen drivers. a. Plot the data as a scatter plot. b. Fit a line to this data.
29 Chapter 1. Polynomials and Factoring; More on Probability c. Predict the number of teenage traffic accidents attributable to cell phones in the year
30 1.5. Factoring Quadratic Expressions TABLE 1.1: Year (y) Total (n) Simplify Graph the following on a number line: π, 2, 5 3, 3 10, What is the multiplicative inverse of 9 4? Quick Quiz 1. Name the following polynomial. State its degree and leading coefficient 6x 2 y 4 z + 6x 6 2y xyz Simplify (a 2 b 2 c + 11abc 5 ) + (4abc 5 3a 2 b 2 c + 9abc). 3. A rectangular solid has dimensions (a + 2) by (a + 4) by (3a). Find its volume. 4. Simplify 3h jk 3 (h 2 j 4 k + 6hk 2 ). 5. Find the solutions to (x 3)(x + 4)(2x 1) = Multiply (a 9b)(a + 9b). 28
31 Chapter 1. Polynomials and Factoring; More on Probability 1.6 Factoring Special Products When we learned how to multiply binomials, we talked about two special products: the Sum and Difference Formula and the Square of a Binomial Formula. In this lesson, we will learn how to recognize and factor these special products. Factoring the Difference of Two Squares We use the Sum and Difference Formula to factor a difference of two squares. A difference of two squares can be a quadratic polynomial in this form: a 2 b 2. Both terms in the polynomial are perfect squares. In a case like this, the polynomial factors into the sum and difference of the square root of each term. a 2 b 2 = (a + b)(a b) In these problems, the key is figuring out what the a and b terms are. Let s do some examples of this type. Example 1: Factor the difference of squares. (a) x 2 9 (b) x 2 y 2 1 Solution: (a) Rewrite as x 2 9 as x Now it is obvious that it is a difference of squares. We substitute the values of a and b for the Sum and Difference Formula: (x + 3)(x 3) The answer is x 2 9 = (x + 3)(x 3). (b) Rewrite as x 2 y 2 1 as (xy) This factors as (xy + 1)(xy 1). Factoring Perfect Square Trinomials A perfect square trinomial has the form a 2 + 2ab + b 2 or a 2 2ab + b 2 The factored form of a perfect square trinomial has the form 29
32 1.6. Factoring Special Products And (a + b) 2 i f a 2 + 2(ab) + b 2 (a b) 2 i f a 2 2(ab) + b 2 In these problems, the key is figuring out what the a and b terms are. Let s do some examples of this type. Example: x 2 + 8x + 16 Solution: Check that the first term and the last term are perfect squares. x 2 + 8x + 16 as x 2 + 8x Check that the middle term is twice the product of the square roots of the first and the last terms. This is true also since we can rewrite them. x 2 + 8x + 16 as x x This means we can factor x 2 + 8x + 16 as (x + 4) 2. Example 2: Factor x 2 4x + 4. Solution: Rewrite x 2 4x + 4 as x ( 2) x + ( 2) 2. We notice that this is a perfect square trinomial and we can factor it as: (x 2) 2. Solving Polynomial Equations Involving Special Products We have learned how to factor quadratic polynomials that are helpful in solving polynomial equations like ax 2 + bx + c = 0. Remember that to solve polynomials in expanded form, we use the following steps: Step 1: Rewrite the equation in standard form such that: Polynomial expression = 0. Step 2: Factor the polynomial completely. Step 3: Use the Zero Product Property to set each factor equal to zero. Step 4: Solve each equation from step 3. Step 5: Check your answers by substituting your solutions into the original equation. Example 3: Solve the following polynomial equations. x 2 + 7x + 6 = 0 Solution: No need to rewrite because it is already in the correct form. Factor: We write 6 as a product of the following numbers: 6 = 6 1 and = 7 x 2 + 7x + 6 = 0 factors as (x + 1)(x + 6) = 0 30
33 Chapter 1. Polynomials and Factoring; More on Probability Set each factor equal to zero: x + 1 = 0 or x + 6 = 0 Solve: x = 1 or x = 6 Check: Substitute each solution back into the original equation. ( 1) 2 + 7( 1) + 6 = 1 + ( 7) + 6 = 0 ( 6) 2 + 7( 6) + 6 = 36 + ( 42) + 6 = 0 Practice Set Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra:Factoring Special Products (10:08) MEDIA Click image to the left for more content. Factor the following perfect square trinomials. 1. x 2 + 8x x 2 18x x x x x x 2 4x x x x 2 12xy + 9y 2 8. x x Factor the following difference of squares. 9. x x x
34 1.6. Factoring Special Products x x x x x 2 81y 2 Solve the following quadratic equations using factoring. 17. x 2 11x + 30 = x 2 + 4x = x = 14x 20. x 2 64 = x 2 24x = x 2 25 = x x = x 2 16x 60 = 0 Mixed Review 25. Find the value for k that creates an infinite number of solutions to the system { 3x + 7y = 1 kx 14y = A restaurant has two kinds of rice, three choices of mein, and four kinds of sauce. How many plate combinations can be created if you choose one of each? 27. Graph y 5 = 1 3 (x + 4). Identify its slope. 28. $600 was deposited into an account earning 8% interest compounded annually. a. Write the exponential model to represent this situation. b. How much money will be in the account after six years? 29. Divide Identify an integer than is even and not a natural number. 32
35 Chapter 1. Polynomials and Factoring; More on Probability 1.7 Factoring Polynomials Completely We say that a polynomial is factored completely when we factor as much as we can and we are unable to factor any more. Here are some suggestions that you should follow to make sure that you factor completely. Factor all common monomials first. Identify special products such as difference of squares or the square of a binomial. Factor according to their formulas. If there are no special products, factor using the methods we learned in the previous sections. Look at each factor and see if any of these can be factored further. Example 1: Factor the following polynomials completely. (a) 2x 2 8 (b) x 3 + 6x 2 + 9x Solution: (a) Look for the common monomial factor. 2x 2 8 = 2(x 2 4). Recognize x 2 4 as a difference of squares. We factor 2(x 2 4) = 2(x + 2)(x 2). If we look at each factor we see that we can t factor anything else. The answer is 2(x + 2)(x 2). (b) Recognize this as a perfect square and factor as x(x + 3) 2. If we look at each factor we see that we can t factor anything else. The answer is x(x + 3) 2. Factoring Common Binomials The first step in the factoring process is often factoring the common monomials from a polynomial. Sometimes polynomials have common terms that are binomials. For example, consider the following expression. x(3x + 2) 5(3x + 2) You can see that the term (3x + 2) appears in both terms of the polynomial. This common term can be factored by writing it in front of a set of parentheses. Inside the parentheses, we write all the terms that are left over when we divide them by the common factor. (3x + 2)(x 5) This expression is now completely factored. Let s look at some examples. Example 2: Factor 3x(x 1) + 4(x 1). Solution: 3x(x 1) + 4(x 1) has a common binomial of (x 1). When we factor the common binomial, we get (x 1)(3x + 4). 33
36 1.7. Factoring Polynomials Completely Factoring by Grouping It may be possible to factor a polynomial containing four or more terms by factoring common monomials from groups of terms. This method is called factoring by grouping. The following example illustrates how this process works. Example 3: Factor 2x + 2y + ax + ay. Solution: There isn t a common factor for all four terms in this example. However, there is a factor of 2 that is common to the first two terms and there is a factor of a that is common to the last two terms. Factor 2 from the first two terms and factor a from the last two terms. 2x + 2y + ax + ay = 2(x + y) + a(x + y) Now we notice that the binomial (x + y) is common to both terms. We factor the common binomial and get. (x + y)(2 + a) Our polynomial is now factored completely. We know how to factor Quadratic Trinomials (ax 2 +bx+c) where a 1 using methods we have previously learned. To factor a quadratic polynomial where a 1, we follow the following steps. 1. We find the product ac. 2. We look for two numbers that multiply to give ac and add to give b. 3. We rewrite the middle term using the two numbers we just found. 4. We factor the expression by grouping. Let s apply this method to the following examples. Example 4: Factor 3x 2 + 8x + 4 by grouping. Solution: Follow the steps outlined above. ac = 3 4 = 12 The number 12 can be written as a product of two numbers in any of these ways: 12 = 1 12 and = = 2 6 and = 8 This is the correct choice. Rewrite the middle term as: 8x = 2x + 6x, so the problem becomes the following. 3x 2 + 8x + 4 = 3x 2 + 2x + 6x + 4 Factor an x from the first two terms and 2 from the last two terms. 34
37 Chapter 1. Polynomials and Factoring; More on Probability x(3x + 2) + 2(3x + 2) Now factor the common binomial (3x + 2). (3x + 2)(x + 2) Our answer is (3x + 2)(x + 2). In this example, all the coefficients are positive. What happens if the b is negative? Example 5: Factor 6x 2 11x + 4 by grouping. Solution: ac = 6 4 = 24 The number 24 can be written as a product of two numbers in any of these ways. 24 = 1 24 and = = ( 1) ( 24) and ( 1) + ( 24) = = 2 12 and = = ( 2) ( 12) and ( 2) + ( 12) = = 3 8 and = = ( 3) ( 8) and ( 3) + ( 8) = 11 This is the correct choice. Rewrite the middle term as 11x = 3x 8x, so the problem becomes: 6x 2 11x + 4 = 6x 2 3x 8x + 4 Factor by grouping. Factor a 3x from the first two terms and factor 4 from the last two terms. 3x(2x 1) 4(2x 1) Now factor the common binomial (2x 1). Our answer is (2x 1)(3x 4). Solving RealWorld Problems Using Polynomial Equations Now that we know most of the factoring strategies for quadratic polynomials, we can see how these methods apply to solving realworld problems. Example 6: The product of two positive numbers is 60. Find the two numbers if one of the numbers is 4 more than the other. Solution: x = one of the numbers and x + 4 equals the other number. The product of these two numbers equals 60. We can write the equation. 35
38 1.7. Factoring Polynomials Completely x(x + 4) = 60 Write the polynomial in standard form. x 2 + 4x = 60 x 2 + 4x 60 = 0 Factor: 60 = 6 ( 10) and 6 + ( 10) = 4 60 = 6 10 and = 4 This is the correct choice. The expression factors as (x + 10)(x 6) = 0. Solve: x + 10 = 0 x 6 = 0 or x = 10 x = 6 Since we are looking for positive numbers, the answer must be positive. x = 6 for one number, and x + 4 = 10 for the other number. Check: 6 10 = 60 so the answer checks. Practice Set Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra:Factor by Grouping and Factoring Completely (13:57) MEDIA Click image to the left for more content. Factor completely x x c x x 2 70x 4. 6x 2 600
39 Chapter 1. Polynomials and Factoring; More on Probability 5. 5t 2 20t x x n n a 2 14a x x x 2 + 3x Factor by grouping x 2 9x + 10x x 2 35x + x x 2 9x x x x 5x x 3 14x x x x b b 2 3b m 3 + 3m 2 + 4m x 2 + 7x x 2 + 8x a 3 5a 2 + 7a x x xy + 32x + 20y ab + 40a + 6b mn + 12m + 3n jk 8 j 2 + 5k 10 j ab + 64a 21b 56 Solve the following application problems. 28. One leg of a right triangle is seven feet longer than the other leg. The hypotenuse is 13 feet. Find the dimensions of the right triangle. 29. A rectangle has sides of x + 2 and x 1. What value of x gives an area of 108? 30. The product of two positive numbers is 120. Find the two numbers if one numbers is seven more than the other. 31. Framing Warehouse offers a pictureframing service. The cost for framing a picture is made up of two parts. The cost of glass is $1 per square foot. The cost of the frame is $2 per linear foot. If the frame is a square, what size picture can you get framed for $20.00? Mixed Review 32. The area of a square varies directly with its side length. a. Write the general variation equation to model this sentence. b. If the area is 16 square feet when the side length is 4 feet, find the area when s = 1.5 f eet. 33. The surface area is the total amount of surface of a threedimensional figure. The formula for the surface area of a cylinder is SA = 2πr 2 + 2πrh, where r = radius and h = height o f the cylinder. Determine the surface area of a soup can with a radius of 2 inches and a height of 5.5 inches. 34. Factor 25g Solve this polynomial when it equals zero. 35. What is the greatest common factor of 343r 3 t,21t 4, and 63rt 5? 36. Discounts to the hockey game are given to groups with more than 12 people. a. Graph this solution on a number line b. What is the domain of this situation? c. Will a church group with 12 members receive a discount? 37
40 1.8. Probability of Compound Events Probability of Compound Events We begin this lesson with a reminder of probability. The experimental probability is the ratio of the proposed outcome to the number of experiment trials. P(success) = number o f times the event occured total number o f trials o f experiment Probability can be expressed as a percentage, a fraction, a decimal, or a ratio. This lesson will focus on compound events and the formulas used to determine the probability of such events. Compound events are two simple events taken together, usually expressed as A and B. Independent and Dependent Events Example: Suppose you flip a coin and roll a die at the same time. What is the probability you will flip a head and roll a four? These events are independent. Independent events occur when the outcome of one event does not affect the outcome of the second event. Rolling a four has no effect on tossing a head. To find the probability of two independent events, multiply the probability of the first event by the probability of the second event. Solution: P(A and B) = P(A) P(B) P(tossing a head) = 1 2 P(rolling a 4) = 1 6 P(tossing a head AND rolling a 4) = = 1 12 When events depend upon each other, they are called dependent events. Suppose you randomly draw a card from a standard deck then randomly draw a second card without replacing the first. The second probability is now different from the first. To find the probability of two dependent events, multiply the probability of the first event by the probability of the second event, after the first event occurs. P(A and B) = P(A) P(B f ollowing A) 38
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