Question

Divide a⁴-16 by a+2 ​

Answer 1

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

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

Explanation:

Given expression:

`(a^4-16)/(a+2)`

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

`\implies (a^(2 \cdot 2)-4^2)/(a+2)`

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

`\implies (\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 ((a^2+4)(a^2-4))/(a+2)`

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

`\implies ((a^2+4)(a^2-2^2))/(a+2)`

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

`\implies ((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`