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Section 8.2. Factoring Using the Distributive Property. Factor polynomials by using the Distributive Property. Solve quadratic equations of the form ax 2 + bx = 0. factoring. factoring by grouping Zero Products Property roots. Factor by Using the Distributive Property.

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Section 8.2

Factoring Using the

Distributive Property

slide2
Factor polynomials by using the Distributive Property.
  • Solve quadratic equations of the form ax2 + bx = 0.
  • factoring
  • factoring by grouping
  • Zero Products Property
  • roots
slide3
Factor by Using the Distributive Property

In Ch.7, you used the distributive property to multiply a polynomial

by a monomial.

2a(6a + 8) = 2a(6a) + 2a(8)

= 12a² + 16a

You can reverse this process to express a polynomial as the product

of a monomial factor and a polynomial factor.

12a² + 16a = 2a(6a) + 2a(8)

= 2a(6a + 8)

Factoring a polynomial means to find its completely factored form.

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Use the Distributive Property

A. Use the Distributive Property to factor 15x + 25x2.

First, find the GCF of 15x + 25x2.

15x = 3 ● 5 ● x Factor each monomial.

25x2 = 5 ● 5 ● x ● x Circle the common prime factors.

GCF: 5 ● x or 5x

Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF.

15x + 25x2 = 5x(3) + 5x(5 ● x) Rewrite each term using the GCF.

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Use the Distributive Property

= 5x(3) + 5x(5x) Simplify remaining factors.

= 5x(3 + 5x) Distributive Property

Answer: 5x(3 + 5x)

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12xy = 2 ● 2 ● 3 ● x ● y Factor each monomial.

24xy2 = 2 ● 2 ● 2 ● 3 ● x ● y ● y

30x2y4 = 2 ● 3 ● 5 ● x ● x● y ● y ● y ● y

Use the Distributive Property

B. Use the Distributive Property to factor 12xy + 24xy2 – 30x2y4.

Circle the common prime factors.

GCF: 2 ● 3 ● x ● y or 6xy

12xy + 24xy2 – 30x2y4 = 6xy(2) + 6xy(4y) + 6xy(–5xy3) Rewrite each term using the GCF.

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Use the Distributive Property

= 6xy(2 + 4y – 5xy3) Distributive Property

Answer: The factored form of 12xy + 24xy2 – 30x2y4 is 6xy(2 + 4y – 5xy3).

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Using the Distributive Property to factor polynomials having four

or more terms is called factoring by groupingbecause pairs of terms

are grouped together and factored.

slide9
Use Grouping

Factor 2xy + 7x – 2y – 7.

2xy + 7x – 2y – 7

= (2xy – 2y) + (7x – 7) Group terms with common factors.

= 2y(x – 1) + 7(x – 1) Factor the GCF from each grouping.

= (x – 1)(2y + 7) Distributive Property

Answer: (x – 1)(2y + 7)

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Recognizing binomials that are additive inverses is often helpful when
  • factoring by grouping.
  • For example, 7 - y and y – 7 are additive inverses.
  • By rewriting 7 - yas -1(y – 7), factoring by grouping is possible
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Use the Additive Inverse Property

Factor 15a – 3ab + 4b – 20.

15a – 3ab + 4b – 20 = (15a – 3ab) + (4b – 20)Group terms with common factors.

= 3a(5 – b) + 4(b – 5)Factor GCF from each grouping.

= 3a(–1)(b – 5) + 4(b – 5)5 – b = –1(b – 5)

= –3a(b – 5) + 4(b – 5)3a(–1) = –3a

Answer: = (b – 5)(–3a + 4) Distributive Property

slide12
Some equations can be solved by factoring. Consider the following:

6(0) = 0 0(-3) = 0 (5 – 5)(0) = 0 -2(-3 + 3) = 0

Notice that in each case, at least one of the factors is zero.

The solutions of an equation are called the roots of the equation.

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Solve an Equation

A. Solve (x – 2)(4x – 1) = 0. Check the solution.

If (x – 2)(4x – 1) = 0, then according to the Zero Product Property, either x – 2 = 0 or 4x – 1 = 0.

(x – 2)(4x – 1) = 0 Original equation

x – 2 = 0 or 4x – 1 = 0 Set each factor equal to zero.

x = 2 4x = 1 Solve each equation.

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Homework Assignment #43

8.2 Skills Practice Sheet

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