Question

1. find all of the zeros : $f(x) = x^3 + 7x^2 + 25x + 175$

a) $-7i, -5i, 5i$ b) $-5i, -7i, 7i$ c) $-5, -7i, 7i$ d) $-7, -5i, 5i$

Explanation:

Step1: Factor by grouping

Group terms:

$$ f(x) = (x^3 +7x^2) + (25x +175) = x^2(x+7) +25(x+7) = (x^2+25)(x+7) $$

Step2: Solve for zeros

Set $f(x)=0$:

1. $x+7=0 \implies x=-7$
2. $x^2+25=0 \implies x^2=-25 \implies x=\pm 5i$

Answer:

d) $-7,-5i,5i$