Question

1. choose efficient methods what is the completely factored form of the expression $16x^{4}-y^{4}$? describe the method(s) of factoring you used.

Explanation:

Step1: Recognize difference of squares

$16x^4 - y^4 = (4x^2)^2 - (y^2)^2$

Step2: Apply difference of squares rule

$(4x^2)^2 - (y^2)^2 = (4x^2 + y^2)(4x^2 - y^2)$

Step3: Factor remaining difference of squares

$4x^2 - y^2 = (2x)^2 - y^2 = (2x + y)(2x - y)$

Step4: Combine all factors

Substitute back into the expression from Step2.

Answer:

The completely factored form is $(4x^2 + y^2)(2x + y)(2x - y)$. The method used is the difference of squares factoring rule ($a^2 - b^2 = (a+b)(a-b)$), applied twice: first to factor $16x^4 - y^4$ into $(4x^2 + y^2)(4x^2 - y^2)$, then to factor the remaining difference of squares $4x^2 - y^2$ into $(2x + y)(2x - y)$.