15.1 Factoring Polynomials


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1 LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE ACTIVITY Factoring and Greatest Common Factor Factors that are shared by two or more whole numbers are called common factors. The greatest of these common factors is called the greatest common factor, or GCF. The greatest common factor is 4. Use the greatest common factor (GCF) and the Distributive Property to factor the expression 30x A Write out the prime factors of each term. 30x + 18 = 2 x + 2 B Circle the common factors. 30x + 18 = 2 x + 2 C Write the expression as the product of the GCF and a sum. 30x + 18 = ( ) ( x + ) REFLECT 1. Will you get a completely factored expression if you factor out a common factor that is not the GCF? Explain. 2. Is the expression 2(3x  4x) completely factored? Explain. Lesson
2 Greatest Common Factor of Monomials To find the GCF of monomials, factor each coefficient and write all powers of variables as products. Then find the product of the common factors. Math On the Spot EXAMPLE 1 Find the GCF of each pair of monomials. A 3 x 3 and 6 x 2 3x 3 = 3 x x x 6x 2 = 2 3 x x Factor each coefficient and write powers as products. Find the common factors. 3 x x Find the product of the common factors. The GCF of 3x 3 and 6x 2 is 3x 2. B 4x 2 and 5y 3 Math Talk Mathematical Practices Does factoring an expression change its value? 4x 2 = 2 2 x x 5y 3 = 5 y y y Factor each coefficient and write powers as products. Since there are no common factors other than 1, the GCF of 4x 2 and 5y 3 is 1. REFLECT 3. Analyze Relationships If two terms contain the same variable raised to different powers, to what power will the variable be raised in the GCF? 4. Can the GCF of two positive numbers be greater than both numbers? Explain. YOUR TURN Find the GCF of each pair of monomials g 2 and 27g a 6 and 9b g 4 and 45g ab and 16bc 524 Unit 4
3 Factoring by Using the GCF Remember that the Distributive Property states that ab + ac = a(b + c). Use the Distributive Property to factor out the GCF of the terms in a polynomial to write the polynomial in factored form. EXAMPLE 2 Math On the Spot Factor each polynomial. Check your answer. A 10y y 25y 2y 2 (5y) + 4y(5y)  1(5y) The GCF is 5y. 5y( 2y 2 + 4y  1) Use the Distributive Property. My Notes Check: 5y( 2y 2 + 4y  1) 10y y 25y The product is the original polynomial. B 12x  8x 2 Both coefficients are negative. 1(12x + 8x 2 ) Factor out 1. 1[3(4x) + 2x(4x)] The GCF of 12x and 8x 2 is 4x. 1[4x(3 + 2x)] 1(4x)(3 + 2x) Use the Distributive Property. Use the Associative Property. 4x(3 + 2x) Check: 4x(3 + 2x) = 12x  8x 2 The product is the original polynomial. REFLECT 9. Can the polynomial 5x be factored? Explain. YOUR TURN Factor each polynomial. Check your answer y 212 y x x 32 x 2 Lesson
4 Math On the Spot Factoring Out a Common Binomial Factor Sometimes the GCF of the terms in an expression is a binomial. Such a GCF is called a common binomial factor. You factor out a common binomial factor the same way you factor out a monomial factor. EXAMPLE 3 Factor each expression. My Notes A 7(x  3)  2x(x  3) 7(x 3)  2x(x 3) (x  3) is a common binomial factor. (x 3)(72x) Factor out (x  3). B t(t2 + 4) + (t2 + 4) t( t 2 + 4) + ( t 2 + 4) ( t 2 + 4) is a common binomial factor. t( t 2 + 4) + 1( t 2 + 4) ( t 2 + 4) = 1( t 2 + 4) ( t 2 + 4)(t + 1) Factor out ( t 2 + 4). C 5x(x + 3)  4(3 + x) 5x(x + 3)  4(3 + x) 5x(x + 3)  4(x + 3) (3 + x) = (x + 3), so (x + 3) is a common binomial factor. (x + 3)(5x  4) Factor out (x + 3). D 3x2 (x + 2) + 4(x  7) 3x 2 (x + 2) + 4(x  7) There are no common factors. The expression cannot be factored. YOUR TURN Factor each expression, if possible x(2x + 3) + (2x + 3) x(x + 2) + 9(x + 2) 14. 7(3t  2) + 2t2 (2t  3) 15. 5t(t + 6)  8(6 + t) 526 Unit 4
5 Factoring by Grouping Some polynomials can be factored by grouping. When a polynomial has four terms, you may be able to make two groups and factor the GCF from each. EXAMPLE 4 Factor each polynomial by grouping. Check your answer. Math On the Spot A 12a 39a a  15 (12a 39a 2 ) + (20a  15) 3a 2 (4a  3) + 5(4a  3) 3a 2 (4a  3) + 5(4a  3) Group terms that have a common number or variable as a factor. Factor out the GCF of each group. (4a  3) is a common factor. (4a  3)( 3a 2 + 5) Factor out (4a  3). Check: (4a  3)( 3a 2 + 5) Multiply using FOIL. 4a( 3a 2 ) + 4a(5)  3( 3a 2 )  3(5) 12a a  9a a 39a a  15 The product is the original polynomial. B 2g g 3 + g + 5 ( 2g g 3 ) + (g + 5) Group terms. 2g 3 (g + 5) + 1(g + 5) 2g 3 (g + 5) + 1(g + 5) Factor out the GCF of each group. (g + 5) is a common factor. (g + 5)( 2g 3 + 1) Factor out (g + 5). Check: (g + 5)( 2g 3 + 1) Multiply using FOIL. g( 2g 3 ) + g(1) + 5( 2g 3 ) + 5(1) 2g 4 + g + 10g g g 3 + g + 5 The product is the original polynomial. YOUR TURN Factor each polynomial. Check your answer b 3 + 8b 2 + 9b r r + r Lesson
6 Factoring with Opposites Recognizing opposite binomials can help you factor polynomials. The binomials (5  x) and (x  5) are opposites, because (5  x) = 1(x  5). Math On the Spot EXAMPLE 5 Factor the polynomial by grouping and using opposites. Check your answer. My Notes 3x 315x x ( 3x 315x 2 ) + (102x) Group terms. 3x 2 (x  5) + 2(5  x) Factor out the GCF of each group. 3x 2 (x 5) + 2( 1)(x 5) Write (5  x) as 1(x  5). 3x 2 (x 5)  2(x 5) Simplify. (x 5)( 3x 22) Factor out (x  5). Check: (x  5)( 3x 22) Multiply using FOIL. x( 3x 2 )  x(2)  5( 3x 2 )  5(2) 3x 32x  15x x 315x x The product is the original polynomial. REFLECT 18. Critique Reasoning Inara thinks that the opposite of (a  b) is (a + b), since addition and subtraction are opposites. Is she correct? Explain. YOUR TURN Factor each polynomial. Check your answer x 210x 3 + 8x y x + xy n 618n 556n t 448t 33t Unit 4
7 Guided Practice Write the expression as a product of the greatest common factor and a sum. (Explore Activity) 1. 15y y a. Write out the prime factors of each term. 15y3 + 20y = 3 y + 2 y b. Circle the common factors. 15y3 + 20y = 3 y + 2 y c. Write the product of the GCF and a sum. 15y3 + 20y = ( )( y2 + ) Find the GCF of each pair of monomials. (Example 1) 2. 9s and 63s 3 9s = y y 214y 3 = 63s 3 = y 2 = The GCF of 9s and 63s 3 is. The GCF of 14y 3 and 28y 2 is. Factor each polynomial. (Example 2) y 37y 2  y y( y ) 5. 9d 218 ( d 2  ) 6. 6 x 42x x t Factor each expression. (Example 3) 8. 4s(s + 6)  5(s + 6) 9. 3(2 + b) + 4b(b + 2) ( )(s + 6) ( )( ) 10. (6z)(z + 8) + (z + 8) 11. 8w(5  w) + 3(w  5) Lesson
8 Factor each polynomial. (Example 4) 12. 9x x 2 + x m 3 + 4m 2 + 6m + 12 ( 9x 3 + ) + ( ) ( + 4m 2 ) + ( ) (x + ) + ( ) (m + ) + (m + ) ( )( ) (m + )( ) 2(m + )( ) x 340x2 + 14x n 52n4 + 7n27n Factor each polynomial. (Example 5) r 26r r q 221q + 64q (2 r 2  ) + ( ) ( ) + ( ) (  3) + ( ) 7q( ) + 2( ) 2r(r  3) + 4 ( ) 7q( ) + 2 ( ) ( )( ) ( )( ) ( )( )? 18. 6c c 25c x 327x x ESSENTIAL QUESTION CHECKIN 20. How can you use the greatest common factor to factor polynomials? 530 Unit 4
9 Name Class Date 15.1 Independent Practice 21. Find the GCF of 64n4 and 24n 2. Factor each expression or state if it cannot be factored q p, A.SSE n n + 7n 29. After t years, the amount of money in a savings account that earns simple interest is P + Prt, where P is the starting amount and r is the yearly interest rate. Factor this expression. 30. Communicate Mathematical Ideas Explain how you can show that (x a) and (a x) are opposites b(b + 3) + 5(b + 3) 25. 4(x  3)  x (y + 2) 26. 7r 335r2 + 6r Explain how to check that a polynomial has been factored correctly. 31. The solar panel on Mandy s calculator has an area of ( 7x 2 + x) cm 2. Factor this polynomial to find possible expressions for the dimensions of the solar panel. 28. Explain the Error Billie says the factored form of 18 x 8 9 x 4 6 x 3 is 3x(6 x 7 3 x 3 2x 2 ). Explain her error and give the correct factored form. 32. A model rocket is fired vertically into the air at 320 ft/s. The expression 16t t gives the rocket s height after t seconds. Factor this expression. 33. The area of a triangle is 1_ 2 (x 32x + 2 x 24). The height h is x + 2. Write an expression for the base b of the triangle. (Hint: Area of a triangle = 1_ 2 bh) Lesson
10 34. Raspberries come in a container with a square bottom whose bottom side length is x. An expression for its volume is x 3 2x 2. Blueberries come in a container with a square bottom whose bottom side length is (x 2). An expression for its volume is x 3 4x 2 + 4x. Factor both expressions. 35. The area of a rectangle is represented by the polynomial x 2 + 3x 6x 18. a. Find possible expressions for the length and width of the rectangle. b. Use your answers from part a to find the length, width, and area of the rectangle if x = 12. FOCUS ON HIGHER ORDER THINKING Work Area 36. Critical Thinking Show two methods of factoring the expression ax  bx  ay + by. Is the result the same? 37. Explain the Error Audrey and Owen came up with two different answers when they factored the expression 3 n 3 n 2. Who was correct? Explain the error. 38. Communicating Mathematical Ideas Describe how to find the area of the figure. Show each step and write your answer in factored form. Owen Audrey 3 n 3  n 2 3n 3  n 2 n 2 (3n)  n 2 (0) n 2 (3n)  n 2 (1) n 2 (3n  0) n 2 (3n  1) 2x 2x + 6 x + 8 Image Credits: Getty Images/Photodisc 532 Unit 4
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