Question

1. factor (25a^{4}-b^{4}):

a. ((5a^{2}+b^{2})(5a^{2}-b^{2}))
b. ((5a^{2}-b^{2})^{2})
c. ((5a^{2}+3b^{2})(5a^{2}-3b^{2}))
d. ((-5a^{2}+b^{2})(5a^{2}-b^{2}))

Explanation:

Step1: Recognize difference of squares

$25a^4 - b^4 = (5a^2)^2 - (b^2)^2$

Step2: Apply difference of squares formula

Recall $x^2 - y^2=(x+y)(x-y)$. Here $x=5a^2$, $y=b^2$:
$25a^4 - b^4=(5a^2 + b^2)(5a^2 - b^2)$

Step3: Verify other options

• Option b: $(5a^2 - b^2)^2=25a^4-10a^2b^2+b^4

eq25a^4-b^4$

• Option c: $(5a^2+3b^2)(5a^2-3b^2)=25a^4-9b^4

eq25a^4-b^4$

• Option d: $(-5a^2+b^2)(5a^2-b^2)=-25a^4+10a^2b^2-b^4

eq25a^4-b^4$

Answer:

a. $(5a^{2}+b^{2})(5a^{2}-b^{2})$