Question

given $p(x)=x^3 + 5x^2 + 25x + 125$, find the zeros, real and non-real of $p$
the zeros are
write $p$ in factored form as a product of linear factors.
$p(x) = $
question help: video written example

Explanation:

Step1: Factor by grouping

Group terms to factor pairs.
$P(x) = (x^3 + 5x^2) + (25x + 125)$

Step2: Factor out common terms

Extract GCF from each group.
$P(x) = x^2(x + 5) + 25(x + 5)$

Step3: Combine factored groups

Factor out the shared binomial.
$P(x) = (x^2 + 25)(x + 5)$

Step4: Find real zero

Set linear factor to 0 and solve.
$x + 5 = 0 \implies x = -5$

Step5: Find non-real zeros

Set quadratic to 0 and solve.
$x^2 + 25 = 0 \implies x^2 = -25 \implies x = \pm 5i$

Step6: Rewrite as linear factors

Express $x^2+25$ as linear factors with complex numbers.
$x^2 + 25 = (x - 5i)(x + 5i)$

Answer:

The zeros are $\boldsymbol{-5,\ 5i,\ -5i}$
$P(x) = \boldsymbol{(x + 5)(x - 5i)(x + 5i)}$