Algebra Examples

$x^{6} - 4 x^{4} - 8 x^{4} + 32 x^{2} + 16 x^{2} - 64$

Step 1

Regroup terms.

$x^{6} - 8 x^{4} + 16 x^{2} - 4 x^{4} + 32 x^{2} - 64$

Step 2

Factor $x^{2}$ out of $x^{6} - 8 x^{4} + 16 x^{2}$ .

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Step 2.1

Factor $x^{2}$ out of $x^{6}$ .

$x^{2} x^{4} - 8 x^{4} + 16 x^{2} - 4 x^{4} + 32 x^{2} - 64$

Step 2.2

Factor $x^{2}$ out of $- 8 x^{4}$ .

$x^{2} x^{4} + x^{2} (- 8 x^{2}) + 16 x^{2} - 4 x^{4} + 32 x^{2} - 64$

Step 2.3

Factor $x^{2}$ out of $16 x^{2}$ .

$x^{2} x^{4} + x^{2} (- 8 x^{2}) + x^{2} \cdot 16 - 4 x^{4} + 32 x^{2} - 64$

Step 2.4

Factor $x^{2}$ out of $x^{2} x^{4} + x^{2} (- 8 x^{2})$ .

$x^{2} (x^{4} - 8 x^{2}) + x^{2} \cdot 16 - 4 x^{4} + 32 x^{2} - 64$

Step 2.5

Factor $x^{2}$ out of $x^{2} (x^{4} - 8 x^{2}) + x^{2} \cdot 16$ .

$x^{2} (x^{4} - 8 x^{2} + 16) - 4 x^{4} + 32 x^{2} - 64$

Step 3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x^{2} ({(x^{2})}^{2} - 8 x^{2} + 16) - 4 x^{4} + 32 x^{2} - 64$

Step 4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x^{2} (u^{2} - 8 u + 16) - 4 x^{4} + 32 x^{2} - 64$

Step 5

Factor using the perfect square rule.

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Step 5.1

Rewrite $16$ as $4^{2}$ .

$x^{2} (u^{2} - 8 u + 4^{2}) - 4 x^{4} + 32 x^{2} - 64$

Step 5.2

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

$8 u = 2 \cdot u \cdot 4$

Step 5.3

Rewrite the polynomial.

$x^{2} (u^{2} - 2 \cdot u \cdot 4 + 4^{2}) - 4 x^{4} + 32 x^{2} - 64$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 4$ .

$x^{2} {(u - 4)}^{2} - 4 x^{4} + 32 x^{2} - 64$

Step 6

Replace all occurrences of $u$ with $x^{2}$ .

$x^{2} {(x^{2} - 4)}^{2} - 4 x^{4} + 32 x^{2} - 64$

Step 7

Rewrite $4$ as $2^{2}$ .

$x^{2} {(x^{2} - 2^{2})}^{2} - 4 x^{4} + 32 x^{2} - 64$

Step 8

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

$x^{2} {((x + 2) (x - 2))}^{2} - 4 x^{4} + 32 x^{2} - 64$

Step 9

Factor.

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Step 9.1

Apply the product rule to $(x + 2) (x - 2)$ .

$x^{2} ({(x + 2)}^{2} {(x - 2)}^{2}) - 4 x^{4} + 32 x^{2} - 64$

Step 9.2

Remove unnecessary parentheses.

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} - 4 x^{4} + 32 x^{2} - 64$

Step 10

Factor $4$ out of $- 4 x^{4} + 32 x^{2} - 64$ .

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Step 10.1

Factor $4$ out of $- 4 x^{4}$ .

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 (- x^{4}) + 32 x^{2} - 64$

Step 10.2

Factor $4$ out of $32 x^{2}$ .

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 (- x^{4}) + 4 (8 x^{2}) - 64$

Step 10.3

Factor $4$ out of $- 64$ .

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

Step 10.4

Factor $4$ out of $4 (- x^{4}) + 4 (8 x^{2})$ .

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

Step 10.5

Factor $4$ out of $4 (- x^{4} + 8 x^{2}) + 4 (- 16)$ .

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

Step 11

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 12

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 (- u^{2} + 8 u - 16)$

Step 13

Factor by grouping.

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Step 13.1

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

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Step 13.1.1

Factor $8$ out of $8 u$ .

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 (- u^{2} + 8 (u) - 16)$

Step 13.1.2

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

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 (- u^{2} + (4 + 4) u - 16)$

Step 13.1.3

Apply the distributive property.

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 (- u^{2} + 4 u + 4 u - 16)$

Step 13.2

Factor out the greatest common factor from each group.

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Step 13.2.1

Group the first two terms and the last two terms.

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ((- u^{2} + 4 u) + 4 u - 16)$

Step 13.2.2

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

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 (u (- u + 4) - 4 (- u + 4))$

Step 13.3

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

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ((- u + 4) (u - 4))$

Step 14

Replace all occurrences of $u$ with $x^{2}$ .

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ((- x^{2} + 4) (x^{2} - 4))$

Step 15

Rewrite $4$ as $2^{2}$ .

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ((- x^{2} + 2^{2}) (x^{2} - 4))$

Step 16

Reorder $- x^{2}$ and $2^{2}$ .

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ((2^{2} - x^{2}) (x^{2} - 4))$

Step 17

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

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ((2 + x) (2 - x) (x^{2} - 4))$

Step 18

Rewrite $4$ as $2^{2}$ .

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ((2 + x) (2 - x) (x^{2} - 2^{2}))$

Step 19

Factor.

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Step 19.1

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

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ((2 + x) (2 - x) ((x + 2) (x - 2)))$

Step 19.2

Remove unnecessary parentheses.

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ((2 + x) (2 - x) (x + 2) (x - 2))$

Step 20

Factor.

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Step 20.1

Combine exponents.

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Step 20.1.1

Reorder terms.

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ((x + 2) (2 - x) (x + 2) (x - 2))$

Step 20.1.2

Raise $x + 2$ to the power of $1$ .

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

Step 20.1.3

Raise $x + 2$ to the power of $1$ .

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

Step 20.1.4

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

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

Step 20.1.5

Add $1$ and $1$ .

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ({(x + 2)}^{2} (2 - x) (x - 2))$

Step 20.1.6

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 20.1.7

Factor $- 1$ out of $- x$ .

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

Step 20.1.8

Factor $- 1$ out of $- 1 (- 2) - (x)$ .

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

Step 20.1.9

Remove parentheses.

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ({(x + 2)}^{2} \cdot - 1 (- 2 + x) (x - 2))$

Step 20.1.10

Reorder terms.

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ({(x + 2)}^{2} \cdot - 1 (x - 2) (x - 2))$

Step 20.1.11

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

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ({(x + 2)}^{2} \cdot - 1 ({(x - 2)}^{1} (x - 2)))$

Step 20.1.12

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

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ({(x + 2)}^{2} \cdot - 1 ({(x - 2)}^{1} {(x - 2)}^{1}))$

Step 20.1.13

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

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ({(x + 2)}^{2} \cdot - 1 {(x - 2)}^{1 + 1})$

Step 20.1.14

Add $1$ and $1$ .

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 ({(x + 2)}^{2} \cdot - 1 {(x - 2)}^{2})$

Step 20.2

Remove unnecessary parentheses.

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} + 4 {(x + 2)}^{2} \cdot - 1 {(x - 2)}^{2}$

Step 21

Combine exponents.

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Step 21.1

Factor out negative.

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} - (4 {(x + 2)}^{2} {(x - 2)}^{2})$

Step 21.2

Multiply $4$ by $- 1$ .

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} - 4 ({(x + 2)}^{2} {(x - 2)}^{2})$

Step 22

Remove parentheses.

$x^{2} {(x + 2)}^{2} {(x - 2)}^{2} - 4 {(x + 2)}^{2} {(x - 2)}^{2}$

Step 23

Factor ${(x + 2)}^{2} {(x - 2)}^{2}$ out of $x^{2} {(x + 2)}^{2} {(x - 2)}^{2} - 4 {(x + 2)}^{2} {(x - 2)}^{2}$ .

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Step 23.1

Factor ${(x + 2)}^{2} {(x - 2)}^{2}$ out of $x^{2} {(x + 2)}^{2} {(x - 2)}^{2}$ .

${(x + 2)}^{2} {(x - 2)}^{2} (x^{2}) - 4 {(x + 2)}^{2} {(x - 2)}^{2}$

Step 23.2

Factor ${(x + 2)}^{2} {(x - 2)}^{2}$ out of $- 4 {(x + 2)}^{2} {(x - 2)}^{2}$ .

${(x + 2)}^{2} {(x - 2)}^{2} (x^{2}) + {(x + 2)}^{2} {(x - 2)}^{2} (- 4)$

Step 23.3

Factor ${(x + 2)}^{2} {(x - 2)}^{2}$ out of ${(x + 2)}^{2} {(x - 2)}^{2} (x^{2}) + {(x + 2)}^{2} {(x - 2)}^{2} (- 4)$ .

${(x + 2)}^{2} {(x - 2)}^{2} (x^{2} - 4)$

Step 24

Rewrite $4$ as $2^{2}$ .

${(x + 2)}^{2} {(x - 2)}^{2} (x^{2} - 2^{2})$

Step 25

Factor.

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Step 25.1

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

${(x + 2)}^{2} {(x - 2)}^{2} ((x + 2) (x - 2))$

Step 25.2

Remove unnecessary parentheses.

${(x + 2)}^{2} {(x - 2)}^{2} (x + 2) (x - 2)$

Step 26

Combine exponents.

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Step 26.1

Raise $x + 2$ to the power of $1$ .

${(x + 2)}^{1} {(x + 2)}^{2} {(x - 2)}^{2} (x - 2)$

Step 26.2

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

${(x + 2)}^{1 + 2} {(x - 2)}^{2} (x - 2)$

Step 26.3

Add $1$ and $2$ .

${(x + 2)}^{3} {(x - 2)}^{2} (x - 2)$

Step 26.4

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

${(x + 2)}^{3} ({(x - 2)}^{1} {(x - 2)}^{2})$

Step 26.5

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

${(x + 2)}^{3} {(x - 2)}^{1 + 2}$

Step 26.6

Add $1$ and $2$ .

${(x + 2)}^{3} {(x - 2)}^{3}$