Question

which expression is equivalent to $2x^4 - 20x^2 + 18$?
select all that apply.

a $2(x^4 - 10x^2 + 9)$

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

c $(x^2 + 1)(x^2 - 10)$

d $2(x^2 - 1)(x^2 - 9)$

e $2(x^2 + 1)(x + 3)(x - 3)$

f $2(x + 1)(x - 1)(x + 3)(x - 3)$

Explanation:

Step1: Factor out the GCF

$2x^4 - 20x^2 + 18 = 2(x^4 - 10x^2 + 9)$

Step2: Factor the quartic trinomial

Let $y=x^2$, so $x^4 -10x^2 +9 = y^2 -10y +9 = (y-1)(y-9)$. Substitute back $y=x^2$: $(x^2 -1)(x^2 -9)$

Step3: Factor difference of squares

$x^2 -1=(x+1)(x-1)$; $x^2 -9=(x+3)(x-3)$

Step4: Verify each option

• Option A: Matches Step1 result.
• Option B: Expand $(2x^2-2)(x^2+9)=2x^4+18x^2-2x^2-18=2x^4+16x^2-18$, not equivalent.
• Option C: Expand $(x^2+1)(x^2-10)=x^4-10x^2+x^2-10=x^4-9x^2-10$, not equivalent.
• Option D: Matches Step2 result: $2(x^2-1)(x^2-9)$
• Option E: $2(x^2+1)(x+3)(x-3)=2(x^2+1)(x^2-9)=2(x^4-9x^2+x^2-9)=2x^4-16x^2-18$, not equivalent.
• Option F: Matches Step3 result: $2(x+1)(x-1)(x+3)(x-3)$

Answer:

A. $2(x^4 - 10x^2 + 9)$
D. $2(x^2 - 1)(x^2 - 9)$
F. $2(x + 1)(x - 1)(x + 3)(x - 3)$