Question

1. factor $100x^{4}-16y^{4}$:

a. $(5x^{2}-2y^{2})^{2}$
b. $4(5x^{2}+2y^{2})(5x^{2}-2y^{2})$
c. $4(25x^{2}+4y^{2})^{2}$
d. $16(5x^{2}-2y)^{2}$

Explanation:

Step1: Factor out GCF

Identify greatest common factor (GCF) of $100x^4$ and $16y^4$, which is 4.
$100x^4 - 16y^4 = 4(25x^4 - 4y^4)$

Step2: Recognize difference of squares

Rewrite $25x^4 - 4y^4$ as $(5x^2)^2 - (2y^2)^2$, a difference of squares $a^2 - b^2$.

Step3: Apply difference of squares formula

Use $a^2 - b^2 = (a+b)(a-b)$ where $a=5x^2$, $b=2y^2$.
$25x^4 - 4y^4 = (5x^2+2y^2)(5x^2-2y^2)$

Step4: Combine all factors

Substitute back into the expression from Step1.
$100x^4 - 16y^4 = 4(5x^2+2y^2)(5x^2-2y^2)$

Answer:

b. $4(5x^{2}+2y^{2})(5x^{2}-2y^{2})$