Question

solve the polynomial equation by factoring and then using the zero - product principle.
$x^{3}+3x^{2}=25x + 75$

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

$x^3 + 3x^2 - 25x - 75 = 0$

Step2: Group terms for factoring

$(x^3 + 3x^2) + (-25x - 75) = 0$

Step3: Factor out common terms

$x^2(x + 3) - 25(x + 3) = 0$

Step4: Factor out binomial

$(x + 3)(x^2 - 25) = 0$

Step5: Factor difference of squares

$(x + 3)(x - 5)(x + 5) = 0$

Step6: Apply zero-product principle

Set each factor equal to 0:
$x + 3 = 0$, $x - 5 = 0$, $x + 5 = 0$

Step7: Solve for x

$x = -3$, $x = 5$, $x = -5$

Answer:

A. The solution set is {-5, -3, 5}.