Finite Math Examples

$8 a^{4} b^{3} + 27 a^{7}$

Step 1

Factor out the GCF of $a^{4}$ from $8 a^{4} b^{3} + 27 a^{7}$ .

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Step 1.1

Factor out the GCF of $a^{4}$ from each term in the polynomial.

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Step 1.1.1

Factor out the GCF of $a^{4}$ from the expression $8 a^{4} b^{3}$ .

$a^{4} (8 b^{3}) + 27 a^{7}$

Step 1.1.2

Factor out the GCF of $a^{4}$ from the expression $27 a^{7}$ .

$a^{4} (8 b^{3}) + a^{4} (27 a^{3})$

Step 1.2

Since all the terms share a common factor of $a^{4}$ , it can be factored out of each term.

$a^{4} (8 b^{3} + 27 a^{3})$

Step 2

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

$a^{4} ({(2 b)}^{3} + 27 a^{3})$

Step 3

Rewrite $27 a^{3}$ as ${(3 a)}^{3}$ .

$a^{4} ({(2 b)}^{3} + {(3 a)}^{3})$

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 $a = 2 b$ and $b = 3 a$ .

$a^{4} ((2 b + 3 a) ({(2 b)}^{2} - 2 b (3 a) + {(3 a)}^{2}))$

Step 5

Simplify.

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Step 5.1

Apply the product rule to $2 b$ .

$a^{4} ((2 b + 3 a) (2^{2} b^{2} - 2 b (3 a) + {(3 a)}^{2}))$

Step 5.2

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

$a^{4} ((2 b + 3 a) (4 b^{2} - 2 b (3 a) + {(3 a)}^{2}))$

Step 5.3

Multiply $2$ by $- 1$ .

$a^{4} ((2 b + 3 a) (4 b^{2} - 2 b (3 a) + {(3 a)}^{2}))$

Step 5.4

Rewrite using the commutative property of multiplication.

$a^{4} ((2 b + 3 a) (4 b^{2} - 2 \cdot (3 b a) + {(3 a)}^{2}))$

Step 5.5

Multiply $- 2$ by $3$ .

$a^{4} ((2 b + 3 a) (4 b^{2} - 6 b a + {(3 a)}^{2}))$

Step 5.6

Apply the product rule to $3 a$ .

$a^{4} ((2 b + 3 a) (4 b^{2} - 6 b a + 3^{2} a^{2}))$

Step 5.7

Raise $3$ to the power of $2$ .

$a^{4} ((2 b + 3 a) (4 b^{2} - 6 b a + 9 a^{2}))$