Question

determine the quotient and remainder.
$(x^4 - 81) \div (x + 3)$

Explanation:

Step1: Factor the dividend

$x^4 - 81 = (x^2 - 9)(x^2 + 9) = (x - 3)(x + 3)(x^2 + 9)$

Step2: Cancel common factor

$\frac{(x - 3)(x + 3)(x^2 + 9)}{x + 3} = (x - 3)(x^2 + 9)$

Step3: Expand the quotient

$(x - 3)(x^2 + 9) = x^3 - 3x^2 + 9x - 27$

Step4: Verify remainder

Substitute $x=-3$ into $x^4 - 81$: $(-3)^4 - 81 = 81 - 81 = 0$

Answer:

Quotient: $x^3 - 3x^2 + 9x - 27$
Remainder: $0$