Factoring Special Cases in Algebra
1.) What is the factored form of 2x
2
+ 9x + 10?
2.) What is the factored form of 6x
2
– 23x – 4?
Warm Up
Factor perfect-square trinomials and the
differences of two squares
Objective
“Reversing” the rules for multiplying special case
binomials will result in the original factors.
Any trinomial of the form a
2
+ 2ab + b
2
or a
2
-2ab + b
2
is a
perfect square trinomial because it is the result of
squaring a binomial.
For every real number a and b:
a
2
+ 2ab + b
2
= (a + b)(a + b) = (a + b)
2
x
2
+ 8x + 16 = (x + 4)(x + 4) = (x + 4)
2
a
2
– 2ab + b
2
= (a – b)(a – b) = (a – b)
2
4n
2
– 12n + 9 = (2n – 3)(2n – 3) = (2n – 3)
2
Factoring Perfect Square Trinomials
What is the factored form of x
2
– 4x +4?
Example 1 – Factoring a Perfect-Square
Trinomial
What is the factored form of x
2
+ 6x + 9?
Extra Example 1
Suppose the area of a square can be represented
by the expression 9x
2
+ 24x + 16. What is an
expression for the length of one side of the
square?
Example 2 – Factoring to Find a Length
You can factor a difference of squares, a
2
– b
2
, as
(a + b)(a – b)
For all real numbers a and b:
a
2
– b
2
= (a + b)(a – b)
Ex.:
x
2
– 64 = (x + 8)(x – 8)
25x
2
– 36 = (5x + 6)(5x – 6)
Difference of Two Squares
What is the factored form of x
2
– 64?
Example 3 – Factoring a Difference of
Two Squares
What is the factored form of x
2
– 100?
Extra Example 3
What is the factored form of 9x
2
– 25?
Example 4 – Factoring a Difference of
Two Squares
What is the factored form of 25d
2
– 64?
Extra Example 4
What is the factored form of 12x
2
– 3?
Example 5 – Factoring Out a Common
Factor
Pg. 514 – 515 (9 – 17 all, 24 – 32 all, 36 – 41 all)
Assignment
Learn about factoring perfect square trinomials and the differences of two squares in algebra. Explore examples and understand how to factor expressions efficiently using these special cases.
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Presentation Transcript
Warm Up 1.) What is the factored form of 2x2+ 9x + 10? 2.) What is the factored form of 6x2 23x 4?
8.7 Factoring Special Cases
Objective Factor perfect-square trinomials and the differences of two squares
Factoring Perfect Square Trinomials Reversing the rules for multiplying special case binomials will result in the original factors. Any trinomial of the form a2+ 2ab + b2or a2-2ab + b2is a perfect square trinomial because it is the result of squaring a binomial. For every real number a and b: a2+ 2ab + b2= (a + b)(a + b) = (a + b)2 x2+ 8x + 16 = (x + 4)(x + 4) = (x + 4)2 a2 2ab + b2= (a b)(a b) = (a b)2 4n2 12n + 9 = (2n 3)(2n 3) = (2n 3)2
Example 1 Factoring a Perfect-Square Trinomial What is the factored form of x2 4x +4?
Extra Example 1 What is the factored form of x2+ 6x + 9?
Example 2 Factoring to Find a Length Suppose the area of a square can be represented by the expression 9x2+ 24x + 16. What is an expression for the length of one side of the square?
Difference of Two Squares You can factor a difference of squares, a2 b2, as (a + b)(a b) For all real numbers a and b: a2 b2= (a + b)(a b) Ex.: x2 64 = (x + 8)(x 8) 25x2 36 = (5x + 6)(5x 6)
Example 3 Factoring a Difference of Two Squares What is the factored form of x2 64?
Extra Example 3 What is the factored form of x2 100?
Example 4 Factoring a Difference of Two Squares What is the factored form of 9x2 25?
Extra Example 4 What is the factored form of 25d2 64?
Example 5 Factoring Out a Common Factor What is the factored form of 12x2 3?
Assignment Pg. 514 515 (9 17 all, 24 32 all, 36 41 all)
