Factor 125x^3+1+75x^2+15x

$125 x^{3} + 1 + 75 x^{2} + 15 x$

Step 1

Regroup terms.

$125 x^{3} + 1 + 75 x^{2} + 15 x$

Step 2

Rewrite $125 x^{3}$ as ${(5 x)}^{3}$ .

${(5 x)}^{3} + 1 + 75 x^{2} + 15 x$

Step 3

Rewrite $1$ as $1^{3}$ .

${(5 x)}^{3} + 1^{3} + 75 x^{2} + 15 x$

Step 4

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 5 x$ and $b = 1$ .

$(5 x + 1) ({(5 x)}^{2} - (5 x) \cdot 1 + 1^{2}) + 75 x^{2} + 15 x$

Step 5

Simplify.

Apply the product rule to $5 x$ .

$(5 x + 1) (5^{2} x^{2} - (5 x) \cdot 1 + 1^{2}) + 75 x^{2} + 15 x$

Raise $5$ to the power of $2$ .

$(5 x + 1) (25 x^{2} - (5 x) \cdot 1 + 1^{2}) + 75 x^{2} + 15 x$

Multiply $5$ by $- 1$ .

$(5 x + 1) (25 x^{2} - 5 x \cdot 1 + 1^{2}) + 75 x^{2} + 15 x$

Multiply $- 5$ by $1$ .

$(5 x + 1) (25 x^{2} - 5 x + 1^{2}) + 75 x^{2} + 15 x$

One to any power is one.

$(5 x + 1) (25 x^{2} - 5 x + 1) + 75 x^{2} + 15 x$

Step 6

Factor $15 x$ out of $75 x^{2} + 15 x$ .

Factor $15 x$ out of $75 x^{2}$ .

$(5 x + 1) (25 x^{2} - 5 x + 1) + 15 x (5 x) + 15 x$

Factor $15 x$ out of $15 x$ .

$(5 x + 1) (25 x^{2} - 5 x + 1) + 15 x (5 x) + 15 x (1)$

Factor $15 x$ out of $15 x (5 x) + 15 x (1)$ .

$(5 x + 1) (25 x^{2} - 5 x + 1) + 15 x (5 x + 1)$

Step 7

Factor $5 x + 1$ out of $(5 x + 1) (25 x^{2} - 5 x + 1) + 15 x (5 x + 1)$ .

Factor $5 x + 1$ out of $15 x (5 x + 1)$ .

$(5 x + 1) (25 x^{2} - 5 x + 1) + (5 x + 1) (15 x)$

Factor $5 x + 1$ out of $(5 x + 1) (25 x^{2} - 5 x + 1) + (5 x + 1) (15 x)$ .

$(5 x + 1) (25 x^{2} - 5 x + 1 + 15 x)$

Step 8

Add $- 5 x$ and $15 x$ .

$(5 x + 1) (25 x^{2} + 10 x + 1)$

Step 9

Factor using the perfect square rule.

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

$(5 x + 1) ({(5 x)}^{2} + 10 x + 1)$

Rewrite $1$ as $1^{2}$ .

$(5 x + 1) ({(5 x)}^{2} + 10 x + 1^{2})$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot (5 x) \cdot 1$

Rewrite the polynomial.

$(5 x + 1) ({(5 x)}^{2} + 2 \cdot (5 x) \cdot 1 + 1^{2})$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 5 x$ and $b = 1$ .

$(5 x + 1) {(5 x + 1)}^{2}$

Factor 125x^3+1+75x^2+15x

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