Algebra Examples

$3 a^{3} - 3 a^{2} b + 9 a b^{2} - a^{2} + a b - 3 b^{2}$

Step 1

Regroup terms.

$3 a^{3} - a^{2} - 3 a^{2} b + 9 a b^{2} + a b - 3 b^{2}$

Step 2

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

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Step 2.1

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

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

Step 2.2

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

$a^{2} (3 a) + a^{2} \cdot - 1 - 3 a^{2} b + 9 a b^{2} + a b - 3 b^{2}$

Step 2.3

Factor $a^{2}$ out of $a^{2} (3 a) + a^{2} \cdot - 1$ .

$a^{2} (3 a - 1) - 3 a^{2} b + 9 a b^{2} + a b - 3 b^{2}$

Step 3

Factor $b$ out of $- 3 a^{2} b + 9 a b^{2} + a b - 3 b^{2}$ .

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Step 3.1

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

$a^{2} (3 a - 1) + b (- 3 a^{2}) + 9 a b^{2} + a b - 3 b^{2}$

Step 3.2

Factor $b$ out of $9 a b^{2}$ .

$a^{2} (3 a - 1) + b (- 3 a^{2}) + b (9 a b) + a b - 3 b^{2}$

Step 3.3

Factor $b$ out of $a b$ .

$a^{2} (3 a - 1) + b (- 3 a^{2}) + b (9 a b) + b a - 3 b^{2}$

Step 3.4

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

$a^{2} (3 a - 1) + b (- 3 a^{2}) + b (9 a b) + b a + b (- 3 b)$

Step 3.5

Factor $b$ out of $b (- 3 a^{2}) + b (9 a b)$ .

$a^{2} (3 a - 1) + b (- 3 a^{2} + 9 a b) + b a + b (- 3 b)$

Step 3.6

Factor $b$ out of $b (- 3 a^{2} + 9 a b) + b a$ .

$a^{2} (3 a - 1) + b (- 3 a^{2} + 9 a b + a) + b (- 3 b)$

Step 3.7

Factor $b$ out of $b (- 3 a^{2} + 9 a b + a) + b (- 3 b)$ .

$a^{2} (3 a - 1) + b (- 3 a^{2} + 9 a b + a - 3 b)$

Step 4

Factor out the greatest common factor from each group.

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Step 4.1

Group the first two terms and the last two terms.

$a^{2} (3 a - 1) + b ((- 3 a^{2} + 9 a b) + a - 3 b)$

Step 4.2

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

$a^{2} (3 a - 1) + b (3 a (- a + 3 b) - 1 (- a + 3 b))$

Step 5

Factor.

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Step 5.1

Factor the polynomial by factoring out the greatest common factor, $- a + 3 b$ .

$a^{2} (3 a - 1) + b ((- a + 3 b) (3 a - 1))$

Step 5.2

Remove unnecessary parentheses.

$a^{2} (3 a - 1) + b (- a + 3 b) (3 a - 1)$

Step 6

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

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Step 6.1

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

$(3 a - 1) a^{2} + b (- a + 3 b) (3 a - 1)$

Step 6.2

Factor $3 a - 1$ out of $b (- a + 3 b) (3 a - 1)$ .

$(3 a - 1) a^{2} + (3 a - 1) (b (- a + 3 b))$

Step 6.3

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

$(3 a - 1) (a^{2} + b (- a + 3 b))$

Step 7

Apply the distributive property.

$(3 a - 1) (a^{2} + b (- a) + b (3 b))$

Step 8

Rewrite using the commutative property of multiplication.

$(3 a - 1) (a^{2} - b a + b (3 b))$

Step 9

Rewrite using the commutative property of multiplication.

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

Step 10

Multiply $b$ by $b$ by adding the exponents.

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Step 10.1

Move $b$ .

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

Step 10.2

Multiply $b$ by $b$ .

$(3 a - 1) (a^{2} - b a + 3 b^{2})$