Question

factor ( 75x^4 - 12 ):
a. ( 3(5x^2 + 2)(5x^2 - 2) )
b. ( 3(5x^2 + 1)(5x^2 - 1) )
c. ( (5x^2 - 2)^2 )
d. ( 3(x^2 + 5)(x^2 - 5) )

factor ( 18x^4 - 2 ):
a. ( 2(3x^2 + 2)(3x^2 - 2) )
b. ( 2(3x^2 + 1)(3x^2 - 1) )
c. ( 2(3x^2 + 4)(3x^2 - 4) )
d. ( 2(9x^2 + 1)^2 )

Explanation:

First Problem: Factor $75x^4 - 12$

Step1: Factor out GCF

The greatest common factor of $75x^4$ and $12$ is $3$.
$75x^4 - 12 = 3(25x^4 - 4)$

Step2: Recognize difference of squares

$25x^4 - 4$ is a difference of squares: $a^2 - b^2$ where $a=5x^2$, $b=2$.
$25x^4 - 4 = (5x^2)^2 - (2)^2$

Step3: Apply difference of squares rule

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

Step4: Combine factors

Substitute back into the factored form.
$75x^4 - 12 = 3(5x^2 + 2)(5x^2 - 2)$

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Second Problem: Factor $18x^4 - 2$

Step1: Factor out GCF

The greatest common factor of $18x^4$ and $2$ is $2$.
$18x^4 - 2 = 2(9x^4 - 1)$

Step2: Recognize difference of squares

$9x^4 - 1$ is a difference of squares: $a^2 - b^2$ where $a=3x^2$, $b=1$.
$9x^4 - 1 = (3x^2)^2 - (1)^2$

Step3: Apply difference of squares rule

Use $a^2 - b^2=(a+b)(a-b)$.
$9x^4 - 1 = (3x^2 + 1)(3x^2 - 1)$

Step4: Combine factors

Substitute back into the factored form.
$18x^4 - 2 = 2(3x^2 + 1)(3x^2 - 1)$

Answer:

a. $3(5x^2 + 2)(5x^2 - 2)$