Question

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

Move all terms to one - side: $x^{3}+x^{2}-25x - 25 = 0$.

Step2: Group the terms

Group the first two and the last two terms: $(x^{3}+x^{2})-(25x + 25)=0$.

Step3: Factor out the common factors from each group

Factor out $x^{2}$ from the first group and 25 from the second group: $x^{2}(x + 1)-25(x + 1)=0$.

Step4: Factor out the common binomial factor

Factor out $(x + 1)$: $(x + 1)(x^{2}-25)=0$.

Step5: Factor the difference of squares

Since $x^{2}-25=(x + 5)(x - 5)$ (using the formula $a^{2}-b^{2}=(a + b)(a - b)$ with $a=x$ and $b = 5$), the factored form is $(x + 1)(x + 5)(x - 5)=0$.

Step6: Apply the zero - product principle

Set each factor equal to zero:
If $x+1=0$, then $x=-1$;
If $x + 5=0$, then $x=-5$;
If $x-5=0$, then $x = 5$.

Answer:

A. The solution set is $\{-5,-1,5\}$