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Factor polynomials by using the Distributive Property.

- Solve quadratic equations of the form ax2 + bx = 0.

- factoring

- factoring by grouping
- Zero Products Property
- roots

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.

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.

Use the Distributive Property

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

= 5x(3 + 5x) Distributive Property

Answer: 5x(3 + 5x)

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.

Use the Distributive Property

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

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

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.

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)

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

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

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.

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.

Homework Assignment #43

8.2 Skills Practice Sheet

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