Section 8.2Factoring Using the Distributive Property

• factoring• factoring by grouping• Zero Products Property• roots

• Factor polynomials by using the Distributive Property.

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

Factor by Using the Distributive Property

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

2a(6a + 8) = 2a(6a) + 2a(8) = 12a² + 16a

You can reverse this process to express a polynomial as the productof 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 5xWrite 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)

Use the Distributive Property

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

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

GCF: 2 ● 3 ● x ● y or 6xy

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

Circle the common prime factors.

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 grouping because pairs of termsare 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 whenfactoring by grouping.

• For example, 7 - y and y – 7 are additive inverses.

• By rewriting 7 - y as -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.

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

Solve an Equation

Homework Assignment #43

8.2 Skills Practice Sheet