Basic Math Examples

$6 a^{3} b^{3} - 11 a^{2} b^{2} - 7 a b$

Step 1

Factor $a b$ out of $6 a^{3} b^{3} - 11 a^{2} b^{2} - 7 a b$ .

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Step 1.1

Factor $a b$ out of $6 a^{3} b^{3}$ .

$a b (6 a^{2} b^{2}) - 11 a^{2} b^{2} - 7 a b$

Step 1.2

Factor $a b$ out of $- 11 a^{2} b^{2}$ .

$a b (6 a^{2} b^{2}) + a b (- 11 a b) - 7 a b$

Step 1.3

Factor $a b$ out of $- 7 a b$ .

$a b (6 a^{2} b^{2}) + a b (- 11 a b) + a b (- 7)$

Step 1.4

Factor $a b$ out of $a b (6 a^{2} b^{2}) + a b (- 11 a b)$ .

$a b (6 a^{2} b^{2} - 11 a b) + a b (- 7)$

Step 1.5

Factor $a b$ out of $a b (6 a^{2} b^{2} - 11 a b) + a b (- 7)$ .

$a b (6 a^{2} b^{2} - 11 a b - 7)$

Step 2

Rewrite $a^{2} b^{2}$ as ${(a b)}^{2}$ .

$a b (6 {(a b)}^{2} - 11 (a b) - 7)$

Step 3

Let $u = a b$ . Substitute $u$ for all occurrences of $a b$ .

$a b (6 u^{2} - 11 u - 7)$

Step 4

Factor by grouping.

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Step 4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 6 \cdot - 7 = - 42$ and whose sum is $b = - 11$ .

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Step 4.1.1

Factor $- 11$ out of $- 11 u$ .

$a b (6 u^{2} - 11 (u) - 7)$

Step 4.1.2

Rewrite $- 11$ as $3$ plus $- 14$

$a b (6 u^{2} + (3 - 14) u - 7)$

Step 4.1.3

Apply the distributive property.

$a b (6 u^{2} + 3 u - 14 u - 7)$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$a b ((6 u^{2} + 3 u) - 14 u - 7)$

Step 4.2.2

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

$a b (3 u (2 u + 1) - 7 (2 u + 1))$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$a b ((2 u + 1) (3 u - 7))$

Step 5

Replace all occurrences of $u$ with $a b$ .

$a b ((2 (a b) + 1) (3 (a b) - 7))$

Step 6

Factor.

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Step 6.1

Simplify.

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Step 6.1.1

Remove parentheses.

$a b ((2 a b + 1) (3 (a b) - 7))$

Step 6.1.2

Remove parentheses.

$a b ((2 a b + 1) (3 a b - 7))$

Step 6.2

Remove unnecessary parentheses.

$a b (2 a b + 1) (3 a b - 7)$