Algebra Examples

$x^{3} + 6 x^{2} y + 12 x y^{2} + 8 y^{3}$

Step 1

Regroup terms.

$x^{3} + 8 y^{3} + 6 x^{2} y + 12 x y^{2}$

Step 2

Rewrite $8 y^{3}$ as ${(2 y)}^{3}$ .

$x^{3} + {(2 y)}^{3} + 6 x^{2} y + 12 x y^{2}$

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 $b = 2 y$ .

$(x + 2 y) (x^{2} - x (2 y) + {(2 y)}^{2}) + 6 x^{2} y + 12 x y^{2}$

Step 4

Simplify.

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Step 4.1

Rewrite using the commutative property of multiplication.

$(x + 2 y) (x^{2} - 1 \cdot 2 x y + {(2 y)}^{2}) + 6 x^{2} y + 12 x y^{2}$

Step 4.2

Multiply $- 1$ by $2$ .

$(x + 2 y) (x^{2} - 2 x y + {(2 y)}^{2}) + 6 x^{2} y + 12 x y^{2}$

Step 4.3

Apply the product rule to $2 y$ .

$(x + 2 y) (x^{2} - 2 x y + 2^{2} y^{2}) + 6 x^{2} y + 12 x y^{2}$

Step 4.4

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

$(x + 2 y) (x^{2} - 2 x y + 4 y^{2}) + 6 x^{2} y + 12 x y^{2}$

Step 5

Factor $6 x y$ out of $6 x^{2} y + 12 x y^{2}$ .

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Step 5.1

Factor $6 x y$ out of $6 x^{2} y$ .

$(x + 2 y) (x^{2} - 2 x y + 4 y^{2}) + 6 x y (x) + 12 x y^{2}$

Step 5.2

Factor $6 x y$ out of $12 x y^{2}$ .

$(x + 2 y) (x^{2} - 2 x y + 4 y^{2}) + 6 x y (x) + 6 x y (2 y)$

Step 5.3

Factor $6 x y$ out of $6 x y (x) + 6 x y (2 y)$ .

$(x + 2 y) (x^{2} - 2 x y + 4 y^{2}) + 6 x y (x + 2 y)$

Step 6

Factor $x + 2 y$ out of $(x + 2 y) (x^{2} - 2 x y + 4 y^{2}) + 6 x y (x + 2 y)$ .

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Step 6.1

Factor $x + 2 y$ out of $6 x y (x + 2 y)$ .

$(x + 2 y) (x^{2} - 2 x y + 4 y^{2}) + (x + 2 y) (6 x y)$

Step 6.2

Factor $x + 2 y$ out of $(x + 2 y) (x^{2} - 2 x y + 4 y^{2}) + (x + 2 y) (6 x y)$ .

$(x + 2 y) (x^{2} - 2 x y + 4 y^{2} + 6 x y)$

Step 7

Add $- 2 x y$ and $6 x y$ .

$(x + 2 y) (x^{2} + 4 y^{2} + 4 x y)$

Step 8

Factor using the perfect square rule.

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Step 8.1

Rearrange terms.

$(x + 2 y) (x^{2} + 4 x y + 4 y^{2})$

Step 8.2

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

$(x + 2 y) (x^{2} + 4 x y + {(2 y)}^{2})$

Step 8.3

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

$4 x y = 2 \cdot x \cdot (2 y)$

Step 8.4

Rewrite the polynomial.

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

Step 8.5

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

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