Question

solve the polynomial equation by factoring and then using the zero - product principle.
$x^{3}+x^{2}=81x + 81$
find the solution set. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is { }
(use a comma to separate answers as needed. simplify your answer.)
b. there is no solution.

Explanation:

Step1: Rearrange all terms to left

$x^3 + x^2 - 81x - 81 = 0$

Step2: Factor by grouping

$x^2(x + 1) - 81(x + 1) = 0$

Step3: Factor out common binomial

$(x^2 - 81)(x + 1) = 0$

Step4: Factor difference of squares

$(x - 9)(x + 9)(x + 1) = 0$

Step5: Apply zero-product principle

Set each factor equal to 0:
$x - 9 = 0$, $x + 9 = 0$, $x + 1 = 0$

Step6: Solve for x

$x = 9$, $x = -9$, $x = -1$

Answer:

A. The solution set is {-9, -1, 9}