Question

which two of the following expressions are equivalent to $16x^4 - 64x^{10}$?
$\square$ $(4x^3 - 8x^4)(4x^3 + 8x^4)$
$\square$ $(4x^3 - 8x^4)(4x^3 - 8x^4)$
$\square$ $16x^4(1 - 2x^3)(1 + 2x^3)$
$\square$ $4x^2(4x^2 - 16x^4)$
$\square$ $(2x - 4x^3)(2x + 4x^3)(4x^2 + 8x^4)$

Explanation:

Step1: Factor out GCF

Factor $16x^4$ from $16x^4 - 64x^{10}$:
$16x^4 - 64x^{10} = 16x^4(1 - 4x^6)$

Step2: Apply difference of squares

Rewrite $1-4x^6$ as $1^2-(2x^3)^2$, factor:
$16x^4(1 - 4x^6) = 16x^4(1-2x^3)(1+2x^3)$

Step3: Verify other options

Option1: Expand $(4x^3-8x^4)(4x^3+8x^4)$

$= (4x^3)^2-(8x^4)^2 = 16x^6-64x^8$ (not equivalent)

Option2: Expand $(4x^3-8x^4)^2$

$=16x^6-64x^7+64x^8$ (not equivalent)

Option4: Expand $4x^2(4x^2-16x^4)$

$=16x^4-64x^6$ (not equivalent)

Option5: Expand $(2x-4x^2)(2x+4x^2)(4x^2+8x^4)$

First $(2x-4x^2)(2x+4x^2)=4x^2-16x^4$, then multiply by $4x^2+8x^4$:
$=(4x^2-16x^4)(4x^2+8x^4)=16x^4+32x^6-64x^6-128x^8=16x^4-32x^6-128x^8$ (not equivalent)

Answer:

$\boldsymbol{16x^4(1 - 2x^3)(1 + 2x^3)}$