Write as a Set of Linear Factors 3x^3+5x^2+75x+125

$3 x^{3} + 5 x^{2} + 75 x + 125$

Step 1

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{3} + 5 x^{2}) + 75 x + 125$

Factor out the greatest common factor (GCF) from each group.

$x^{2} (3 x + 5) + 25 (3 x + 5)$

Step 2

Factor the polynomial by factoring out the greatest common factor, $3 x + 5$ .

$(3 x + 5) (x^{2} + 25)$

Step 3

Factor over the complex numbers.

Multiply the constant in the polynomial $x^{2} + 25$ by $- i^{2}$ where $i^{2}$ is equal to $- 1$ .

$(3 x + 5) (x^{2} - 25 i^{2})$

Rewrite $25 i^{2}$ as ${(5 i)}^{2}$ .

$(3 x + 5) (x^{2} - {(5 i)}^{2})$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - i^{2} = (a + i) (a - i)$ where $a = x$ and $i = 5 i$ .

$(3 x + 5) ((x + 5 i) (x - (5 i)))$

Factor.

Multiply $5$ by $- 1$ .

$(3 x + 5) ((x + 5 i) (x - 5 i))$

Remove unnecessary parentheses.

$(3 x + 5) (x + 5 i) (x - 5 i)$