Question

1. factor $9x^4 - 16$:a. $(3x^2 - 4)^2$b. $(3x^2 + 4)(3x^2 - 4)$c. $(x^2 + 5)(x^2 - 5)$d. $(3x^2 + 5)(3x^2 - 5)$2. factor $75a^4 - 3$:a. $3(25a^2 + 1)^2$b. $9(5a^2 - 1)^2$c. $3(5a^2 + 1)(5a^2 - 1)$d. $(5a^2 - 1)^2$3. factor $25a^4 - b^4$:a. $(-5a^2 + b^2)(5a^2 - b^2)$b. $(5a^2 + 3b^2)(5a^2 - 3b^2)$c. $(5a^2 - b^2)^2$d. $(5a^2 + b^2)(5a^2 - b^2)$

Explanation:

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Problem 1: Factor $9x^4 - 16$

Step1: Recognize difference of squares

$9x^4 = (3x^2)^2$, $16=4^2$

Step2: Apply difference of squares rule

$a^2-b^2=(a+b)(a-b)$
$\implies (3x^2)^2 - 4^2=(3x^2+4)(3x^2-4)$
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Problem 2: Factor $75a^4 - 3$

Step1: Factor out greatest common factor

GCF of 75 and 3 is 3
$\implies 3(25a^4 - 1)$

Step2: Recognize difference of squares

$25a^4=(5a^2)^2$, $1=1^2$

Step3: Apply difference of squares rule

$\implies 3[(5a^2)^2-1^2]=3(5a^2+1)(5a^2-1)$
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Problem 3: Factor $25a^4 - b^4$

Step1: Recognize difference of squares

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

Step2: Apply difference of squares rule

$\implies (5a^2)^2-(b^2)^2=(5a^2+b^2)(5a^2-b^2)$
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Answer:

1. b. $(3x^2 + 4)(3x^2 - 4)$
2. c. $3(5a^2 + 1)(5a^2 - 1)$
3. d. $(5a^2 + b^2)(5a^2 - b^2)$