Download 10.7 Factoring using the distributive property

10.7 & 10.8 FACTORING
USING THE
DISTRIBUTIVE
PROPERTY
10.7
Factoring
Special
Products
VOCABULARY
Factor – all numbers and variables in a
mathematical expression
GCF (Greatest Common Factor) – The largest
factor that divides two or more numbers
Distributive Property – multiplying an outside
factor with all factors inside of grouping
symbols
Factoring – the process of separating an
equation into its component parts
VOCABULARY (CONT)
Monomial – an algebraic expression
with only one term
Polynomial – an algebraic
expression with two or more terms
INTRODUCTION
We just reviewed how to use the
distributive property. You can also
reverse the process and express a
polynomial in factored form by using
the distributive property.
YEAH, SO WHAT DOES THAT
MEAN?
Multiplying Polynomials:
• 3(a + b) =
3a + 3b
• x(y – z) =
xy – xz
• 3y(4x + 2) = 3y(4x) + 3y(2)
12xy + 6y
REVERSING THE PROCESS
Factoring polynomials:
3a + 3b
*find the common factor(s) and
remove it from the problem
*write the factor outside of
parentheses and rewrite the rest as
it was in the original
3(a + b)
You Try!
xy – xz
12xy + 6y
EX. 1: USE THE DISTRIBUTIVE
PROPERTY TO FACTOR 10Y 2 + 15Y
First, find the
greatest common
factor for 10y 2 and
15y
10 y 2  2  5  y  y
15 y  3  5  y
The GCF is 5y.
Then, express each
term as the product
of the GCF and its
remaining factors.
10y2 + 15y = 5y(2y + 3)
EX. 2: FACTOR
21AB 2 – 33A 2 BC
EX. 3: FACTOR
6X 3 Y 2 + 14X 2 Y + 2X 2
FACTORING SPECIAL
PRODUCTS
USE THE PATTERNS!
First and last terms are perfect squares!
(2x + 3)2
4x² + 12x + 9
Perfect Square Trinomial!
The middle term is twice the product of the square roots
of the first and third terms.
(2p - 4) (2p + 4) 4p² - 16
The difference of…
Difference of two squares (DTS)!
two squares!
First and last terms are perfect squares!
(2x - y)2
4x² - 4xy + y²
Perfect Square Trinomial!
The middle term is twice the product of the square roots
of the first and third terms.
The key is to recognize when you see a perfect square trinomial or a DTS!
FACTORING PATTERNS!
First and last terms are perfect squares!
a² + 2ab + b2
Perfect Square Trinomial!
(a + b)2
The middle term is twice the product of the square roots
of the first and third terms.
a² - b2
The difference of…
Difference of two squares (DTS)!
(a - b)(a + b)
two squares!
First and last terms are perfect squares!
a² - 2ab + b²
Perfect Square Trinomial!
(a - b)2
The middle term is twice the product of the square roots
of the first and third terms.
The key is to recognize when you see a perfect square trinomial or a DTS!
Factoring Strategy
I. GCF: Always check for the GCF first, no matter what.
II. Binomials:
III. Trinomials:
a.
b. Trial and error:
c. Perfect square trinomial:
FACTOR!
2x²- 18
2(x²- 9)
2(x + 3)(x – 3)
49t²- ¼r2
(7t + ½r)(7t – ½r)
81x²- 25y²
(9x – 5y)(9x + 5y)
27x²- 12
3(9x²- 4)
3(3x + 2)(3x – 2)
DTS!
DTS!
DTS!
DTS!
FACTOR!
-3x²- 18x - 27
-3(x²+ 6x + 9)
Perfect Square Trinomial!
9y²- 60y + 100
-3(x + 3)2
(3y – 10)2
Perfect Square Trinomial!
2x²- 12x + 18
2(x²- 6x + 9)
2(x – 3)2
49x²+ 84x + 36
(7x + 6)2
Perfect Square Trinomial!
Perfect Square Trinomial!
SOLVE!
Divide each side by
3!
3x²- 30x = -75
3x²- 30x + 75 = 0
x²- 10x + 25 = 0
(x – 5)2 = 0
x= 5
Perfect Square Trinomial!
36y²- 121 = 0
DTS!
-6x²+ 8x + 14 = 0
3x²- 4x – 7 = 0
Divide each side by -2!
(6y + 11)(6y – 11) = 0
y = 11/6, -11/6
(x + 1 )(3x - 7) = 0
x = -1, 7/3
SOLVE!
4x²- 1 = 0
DTS!
7x²- 10x = -3
7x²- 10x + 3 = 0
32x²- 80x + 50 = 0
16x²- 40x + 25 = 0
Divide each side by 2!
Perfect Square Trinomial!
(2x + 1)(2x – 1) = 0
x = ½, -½
(7x – 3 )(x – 1) = 0
x = 1, 3/7
(4x – 5)2 = 0
x = 5/4
ASSIGNMENT
10.7 w/s