Question

2-98.
factor each of the following expressions. look for the difference of squares and common factors.
a. $4x^{2}-9y^{2}$
b. $8x^{3}-2x^{7}$
c. $x^{4}-81y^{4}$
d. $8x^{3}+2x^{7}$
e. did you use a shortcut to factor the expressions in parts (a) through (c)?
if so, describe it. if not, what pattern do you see in these expressions?
how can you use that pattern to factor quickly?

Explanation:

Part a:

Step1: Recognize difference of squares

$4x^2 - 9y^2 = (2x)^2 - (3y)^2$

Step2: Apply difference of squares formula

$a^2 - b^2=(a-b)(a+b)$, so:
$(2x - 3y)(2x + 3y)$

Part b:

Step1: Factor out common term

Common factor is $2x^3$, so:
$8x^3 - 2x^7 = 2x^3(4 - x^4)$

Step2: Recognize inner difference of squares

$4 - x^4 = 2^2 - (x^2)^2$

Step3: Apply difference of squares formula

$2^2 - (x^2)^2=(2 - x^2)(2 + x^2)$

Step4: Combine factors

$2x^3(2 - x^2)(2 + x^2)$

Part c:

Step1: Recognize difference of squares

$x^4 - 81y^4=(x^2)^2 - (9y^2)^2$

Step2: Apply difference of squares formula

$(x^2 - 9y^2)(x^2 + 9y^2)$

Step3: Factor remaining difference of squares

$x^2 - 9y^2=(x - 3y)(x + 3y)$

Step4: Combine factors

$(x - 3y)(x + 3y)(x^2 + 9y^2)$

Part d:

Step1: Factor out common term

Common factor is $2x^3$, so:
$8x^3 + 2x^7 = 2x^3(4 + x^4)$

Part e:

Step1: Identify shortcut for (a)-(c)

All are difference of squares: $a^2-b^2=(a-b)(a+b)$

Step2: Describe pattern

Each expression is a square of a monomial minus square of another monomial.

Step3: Explain quick factoring

Rewrite terms as squares, then apply the difference of squares formula directly.

Answer:

a. $\boldsymbol{(2x - 3y)(2x + 3y)}$
b. $\boldsymbol{2x^3(2 - x^2)(2 + x^2)}$
c. $\boldsymbol{(x - 3y)(x + 3y)(x^2 + 9y^2)}$
d. $\boldsymbol{2x^3(4 + x^4)}$
e. Yes, the shortcut is the difference of squares formula. The pattern in (a)-(c) is each expression is of the form $a^2 - b^2$, where $a$ and $b$ are monomials. To factor quickly, rewrite each term as a perfect square, then apply $a^2 - b^2=(a - b)(a + b)$.