Transcript Document

Section 8.2
Factoring Using the
Distributive Property
• 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)
Use the Distributive Property
B. Use the Distributive Property to factor
12xy + 24xy2 – 30x2y4.
12xy = 2 ● 2 ● 3 ● x ● y
24xy2 = 2 ● 2 ● 2 ● 3 ● x ● y ● y
30x2y4 = 2 ● 3 ● 5 ● x ● x ● y ● y ● y ● y
Factor each
monomial.
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 grouping because 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 - 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.
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