Basic Math Examples

$a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$

Step 1

Regroup terms.

$a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$

Step 2

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 $a = a$ and $b = b$ .

$(a + b) (a^{2} - a b + b^{2}) + 3 a^{2} b + 3 a b^{2}$

Step 3

Factor $3 a b$ out of $3 a^{2} b + 3 a b^{2}$ .

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Step 3.1

Factor $3 a b$ out of $3 a^{2} b$ .

$(a + b) (a^{2} - a b + b^{2}) + 3 a b (a) + 3 a b^{2}$

Step 3.2

Factor $3 a b$ out of $3 a b^{2}$ .

$(a + b) (a^{2} - a b + b^{2}) + 3 a b (a) + 3 a b (b)$

Step 3.3

Factor $3 a b$ out of $3 a b (a) + 3 a b (b)$ .

$(a + b) (a^{2} - a b + b^{2}) + 3 a b (a + b)$

Step 4

Factor $a + b$ out of $(a + b) (a^{2} - a b + b^{2}) + 3 a b (a + b)$ .

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Step 4.1

Factor $a + b$ out of $3 a b (a + b)$ .

$(a + b) (a^{2} - a b + b^{2}) + (a + b) (3 a b)$

Step 4.2

Factor $a + b$ out of $(a + b) (a^{2} - a b + b^{2}) + (a + b) (3 a b)$ .

$(a + b) (a^{2} - a b + b^{2} + 3 a b)$

Step 5

Add $- a b$ and $3 a b$ .

$(a + b) (a^{2} + b^{2} + 2 a b)$

Step 6

Factor using the perfect square rule.

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Step 6.1

Rearrange terms.

$(a + b) (a^{2} + 2 a b + b^{2})$

Step 6.2

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

$2 a b = 2 \cdot a \cdot b$

Step 6.3

Rewrite the polynomial.

$(a + b) (a^{2} + 2 \cdot a \cdot b + b^{2})$

Step 6.4

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

$(a + b) {(a + b)}^{2}$