Question

divide a⁴-16 by a+2 ​

Answer 1

Answer:

$\textsf{Factored form}: \quad (a^2+4)(a-2)$

$\textsf{Standard form}: \quad a^3-2a^2+4a-8$

Step-by-step explanation:

Given expression:

$\dfrac{a^4-16}{a+2}$

Rewrite the exponent 4 as 2·2 and 16 as 4²:

$\implies \dfrac{a^{2 \cdot 2}-4^2}{a+2}$

$\textsf{Apply exponent rule} \quad a^{bc}=(a^b)^c$

$\implies \dfrac{\left(a^2\right)^2-4^2}{a+2}$

$\textsf{Apply the Difference of Two Squares Formula} \quad x^2-y^2=\left(x+y\right)\left(x-y\right):$

$\implies \dfrac{(a^2+4)(a^2-4)}{a+2}$

Rewrite 4 in the second parentheses of the numerator as 2²:

$\implies \dfrac{(a^2+4)(a^2-2^2)}{a+2}$

Apply the Difference of Two Squares Formula to (a² - 2²):

$\implies \dfrac{(a^2+4)(a+2)(a-2)}{a+2}$

Cancel the common factor (a + 2):

$\implies (a^2+4)(a-2)$

Expand:

$\implies a^2(a-2)+4(a-2)$

$\implies a^3-2a^2+4a-8$