Algebra Examples

$x^{3} + 3 x^{2} + 3 x + 1$

Step 1

Regroup terms.

$x^{3} + 1 + 3 x^{2} + 3 x$

Step 2

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

$x^{3} + 1^{3} + 3 x^{2} + 3 x$

Step 3

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 $a = x$ and $b = 1$ .

$(x + 1) (x^{2} - x \cdot 1 + 1^{2}) + 3 x^{2} + 3 x$

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $1$ .

$(x + 1) (x^{2} - x + 1^{2}) + 3 x^{2} + 3 x$

Step 4.2

One to any power is one.

$(x + 1) (x^{2} - x + 1) + 3 x^{2} + 3 x$

Step 5

Factor $3 x$ out of $3 x^{2} + 3 x$ .

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Step 5.1

Factor $3 x$ out of $3 x^{2}$ .

$(x + 1) (x^{2} - x + 1) + 3 x (x) + 3 x$

Step 5.2

Factor $3 x$ out of $3 x$ .

$(x + 1) (x^{2} - x + 1) + 3 x (x) + 3 x (1)$

Step 5.3

Factor $3 x$ out of $3 x (x) + 3 x (1)$ .

$(x + 1) (x^{2} - x + 1) + 3 x (x + 1)$

Step 6

Factor $x + 1$ out of $(x + 1) (x^{2} - x + 1) + 3 x (x + 1)$ .

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Step 6.1

Factor $x + 1$ out of $3 x (x + 1)$ .

$(x + 1) (x^{2} - x + 1) + (x + 1) (3 x)$

Step 6.2

Factor $x + 1$ out of $(x + 1) (x^{2} - x + 1) + (x + 1) (3 x)$ .

$(x + 1) (x^{2} - x + 1 + 3 x)$

Step 7

Add $- x$ and $3 x$ .

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

Step 8

Factor using the perfect square rule.

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Step 8.1

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

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

Step 8.2

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

$2 x = 2 \cdot x \cdot 1$

Step 8.3

Rewrite the polynomial.

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

Step 8.4

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

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