Page 560 - Algebra 1

Look for a pattern in the products. (x+1)(x-1)=x2 -1
(x+2)(x-2)=x2 -4 (x+3)(x-3)=x2 -9 (a+b)(a-b)=a2 -b2
(x·x)-1x+1x-(1·1)=x2 -12 (x·x)-2x+2x-(2·2)=x2 -22 (x·x)-3x+3x-(3·3)=x2 -32 (a·a) -ab +ab-(b·b)=a2 -b2
The pattern can be used to factor the difference of two squares.
Factoring the Difference of Two Squares
Determine whether each binomial is the difference of two squares. If so, factor the binomial.
Difference of Two Squares
The factored form of a difference of two squares is:
a2 -b2 =(a+b)(a-b) Example:x2 -49=(x+7)(x-7)
Example
3
a. 4x2-25 SOLUTION
4x2 - 25
= (2 · 2)(x · x) - (5 · 5) = (2x)2 - 52
= (2x + 5)(2x - 5)
b. 9m4-16n6 SOLUTION
9m4 - 16n6
=(3·3)(m2 ·m2)-(4·4)(n3 ·n3) = (3m2)2 - (4n3)2
= (3m2 + 4n3)(3m2 - 4n3)
Factor each term.
Write as a difference of two squares. Factor.
Hint
Use exponent rules. m4 =m2+2 =m2 ·m2 n6 =n3+3 =n3 ·n3
Factoreachterm.
Write as a difference of two squares. Factor.
c. x2-8 SOLUTION
x2 - 8
= (x · x) - (4 · 2) = x2 - 8
d. -64 + z8
SOLUTION
-64 + z8 = z8 - 64
=(z4 ·z4 )-(8·8) = (z4 )2 - 82
= (z4 + 8)(z4 - 8)
Factor each term.
This is not a difference of two squares.
Write terms in descending order. Factoreachterm.
Write as a difference of two squares. Factor.
Lesson 83 545