Factor 1-64a^3-12a+48a^2

$1 - 64 a^{3} - 12 a + 48 a^{2}$

Reorder terms.

$- 64 a^{3} + 48 a^{2} - 12 a + 1$

Rewrite $- 64 a^{3} + 48 a^{2} - 12 a + 1$ in a factored form.

Regroup terms.

$- 64 a^{3} + 1 + 48 a^{2} - 12 a$

Factor $- 1$ out of $- 64 a^{3} + 1$ .

Factor $- 1$ out of $- 64 a^{3}$ .

$- (64 a^{3}) + 1 + 48 a^{2} - 12 a$

Rewrite $1$ as $- 1 (- 1)$ .

$- (64 a^{3}) - 1 (- 1) + 48 a^{2} - 12 a$

Factor $- 1$ out of $- (64 a^{3}) - 1 (- 1)$ .

$- (64 a^{3} - 1) + 48 a^{2} - 12 a$

Rewrite $64 a^{3}$ as ${(4 a)}^{3}$ .

$- ({(4 a)}^{3} - 1) + 48 a^{2} - 12 a$

Rewrite $1$ as $1^{3}$ .

$- ({(4 a)}^{3} - 1^{3}) + 48 a^{2} - 12 a$

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 4 a$ and $b = 1$ .

$- ((4 a - 1) ({(4 a)}^{2} + 4 a \cdot 1 + 1^{2})) + 48 a^{2} - 12 a$

Factor.

Simplify.

Apply the product rule to $4 a$ .

$- ((4 a - 1) (4^{2} a^{2} + 4 a \cdot 1 + 1^{2})) + 48 a^{2} - 12 a$

Raise $4$ to the power of $2$ .

$- ((4 a - 1) (16 a^{2} + 4 a \cdot 1 + 1^{2})) + 48 a^{2} - 12 a$

Multiply $4$ by $1$ .

$- ((4 a - 1) (16 a^{2} + 4 a + 1^{2})) + 48 a^{2} - 12 a$

One to any power is one.

$- ((4 a - 1) (16 a^{2} + 4 a + 1)) + 48 a^{2} - 12 a$

Remove unnecessary parentheses.

$- (4 a - 1) (16 a^{2} + 4 a + 1) + 48 a^{2} - 12 a$

Factor $12 a$ out of $48 a^{2} - 12 a$ .

Factor $12 a$ out of $48 a^{2}$ .

$- (4 a - 1) (16 a^{2} + 4 a + 1) + 12 a (4 a) - 12 a$

Factor $12 a$ out of $- 12 a$ .

$- (4 a - 1) (16 a^{2} + 4 a + 1) + 12 a (4 a) + 12 a (- 1)$

Factor $12 a$ out of $12 a (4 a) + 12 a (- 1)$ .

$- (4 a - 1) (16 a^{2} + 4 a + 1) + 12 a (4 a - 1)$

Factor $4 a - 1$ out of $- (4 a - 1) (16 a^{2} + 4 a + 1) + 12 a (4 a - 1)$ .

Factor $4 a - 1$ out of $- (4 a - 1) (16 a^{2} + 4 a + 1)$ .

$(4 a - 1) (- 1 (16 a^{2} + 4 a + 1)) + 12 a (4 a - 1)$

Factor $4 a - 1$ out of $12 a (4 a - 1)$ .

$(4 a - 1) (- 1 (16 a^{2} + 4 a + 1)) + (4 a - 1) (12 a)$

Factor $4 a - 1$ out of $(4 a - 1) (- 1 (16 a^{2} + 4 a + 1)) + (4 a - 1) (12 a)$ .

$(4 a - 1) (- 1 (16 a^{2} + 4 a + 1) + 12 a)$

Apply the distributive property.

$(4 a - 1) (- 1 (16 a^{2}) - 1 (4 a) - 1 \cdot 1 + 12 a)$

Simplify.

Multiply $16$ by $- 1$ .

$(4 a - 1) (- 16 a^{2} - 1 (4 a) - 1 \cdot 1 + 12 a)$

Multiply $4$ by $- 1$ .

$(4 a - 1) (- 16 a^{2} - 4 a - 1 \cdot 1 + 12 a)$

Multiply $- 1$ by $1$ .

$(4 a - 1) (- 16 a^{2} - 4 a - 1 + 12 a)$

Add $- 4 a$ and $12 a$ .

$(4 a - 1) (- 16 a^{2} + 8 a - 1)$

Factor.

Factor by grouping.

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 $a \cdot c = - 16 \cdot - 1 = 16$ and whose sum is $b = 8$ .

Factor $8$ out of $8 a$ .

$(4 a - 1) (- 16 a^{2} + 8 (a) - 1)$

Rewrite $8$ as $4$ plus $4$

$(4 a - 1) (- 16 a^{2} + (4 + 4) a - 1)$

Apply the distributive property.

$(4 a - 1) (- 16 a^{2} + 4 a + 4 a - 1)$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 a - 1) ((- 16 a^{2} + 4 a) + 4 a - 1)$

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

$(4 a - 1) (4 a (- 4 a + 1) - (- 4 a + 1))$

Factor the polynomial by factoring out the greatest common factor, $- 4 a + 1$ .

$(4 a - 1) ((- 4 a + 1) (4 a - 1))$

Remove unnecessary parentheses.

$(4 a - 1) (- 4 a + 1) (4 a - 1)$

Combine exponents.

Factor $- 1$ out of $4 a$ .

$(- (- 4 a) - 1) (- 4 a + 1) (4 a - 1)$

Rewrite $- 1$ as $- 1 (1)$ .

$(- (- 4 a) - 1 (1)) (- 4 a + 1) (4 a - 1)$

Factor $- 1$ out of $- (- 4 a) - 1 (1)$ .

$- (- 4 a + 1) (- 4 a + 1) (4 a - 1)$

Rewrite $- (- 4 a + 1)$ as $- 1 (- 4 a + 1)$ .

$- 1 (- 4 a + 1) (- 4 a + 1) (4 a - 1)$

Raise $- 4 a + 1$ to the power of $1$ .

$- 1 ({(- 4 a + 1)}^{1} (- 4 a + 1)) (4 a - 1)$

Raise $- 4 a + 1$ to the power of $1$ .

$- 1 ({(- 4 a + 1)}^{1} {(- 4 a + 1)}^{1}) (4 a - 1)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 1 {(- 4 a + 1)}^{1 + 1} (4 a - 1)$

Add $1$ and $1$ .

$- 1 {(- 4 a + 1)}^{2} (4 a - 1)$

Factor $- 1$ out of $- 4 a$ .

$- 1 {(- (4 a) + 1)}^{2} (4 a - 1)$

Rewrite $1$ as $- 1 (- 1)$ .

$- 1 {(- (4 a) - 1 (- 1))}^{2} (4 a - 1)$

Factor $- 1$ out of $- (4 a) - 1 (- 1)$ .

$- 1 {(- (4 a - 1))}^{2} (4 a - 1)$

Rewrite $- (4 a - 1)$ as $- 1 (4 a - 1)$ .

$- 1 {(- 1 (4 a - 1))}^{2} (4 a - 1)$

Apply the product rule to $- 1 (4 a - 1)$ .

$- 1 ({(- 1)}^{2} {(4 a - 1)}^{2}) (4 a - 1)$

Raise $- 1$ to the power of $2$ .

$- 1 (1 {(4 a - 1)}^{2}) (4 a - 1)$

Multiply ${(4 a - 1)}^{2}$ by $1$ .

$- 1 {(4 a - 1)}^{2} (4 a - 1)$

Raise $4 a - 1$ to the power of $1$ .

$- 1 ({(4 a - 1)}^{1} {(4 a - 1)}^{2})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 1 {(4 a - 1)}^{1 + 2}$

Add $1$ and $2$ .

$- 1 {(4 a - 1)}^{3}$

Factor 1-64a^3-12a+48a^2

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