Algebra Examples

$2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x$

Step 1

Factor out the GCF of $2 x$ from $2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x$ .

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Step 1.1

Factor out the GCF of $2 x$ from each term in the polynomial.

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Step 1.1.1

Factor out the GCF of $2 x$ from the expression $2 x^{4}$ .

$2 x (x^{3}) - 4 x^{3} - 8 x^{2} + 16 x$

Step 1.1.2

Factor out the GCF of $2 x$ from the expression $- 4 x^{3}$ .

$2 x (x^{3}) + 2 x (- 2 x^{2}) - 8 x^{2} + 16 x$

Step 1.1.3

Factor out the GCF of $2 x$ from the expression $- 8 x^{2}$ .

$2 x (x^{3}) + 2 x (- 2 x^{2}) + 2 x (- 4 x) + 16 x$

Step 1.1.4

Factor out the GCF of $2 x$ from the expression $16 x$ .

$2 x (x^{3}) + 2 x (- 2 x^{2}) + 2 x (- 4 x) + 2 x (8)$

Step 1.2

Since all the terms share a common factor of $2 x$ , it can be factored out of each term.

$2 x (x^{3} - 2 x^{2} - 4 x + 8)$

Step 2

Factor out the greatest common factor from each group.

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Step 2.1

Group the first two terms and the last two terms.

$2 x ((x^{3} - 2 x^{2}) - 4 x + 8)$

Step 2.2

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

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

Step 3

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

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

Step 4

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

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

Step 5

Factor.

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Step 5.1

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 $b = 2$ .

$2 x ((x - 2) ((x + 2) (x - 2)))$

Step 5.2

Remove unnecessary parentheses.

$2 x ((x - 2) (x + 2) (x - 2))$

Step 6

Combine exponents.

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Step 6.1

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

$2 x ((x - 2) (x - 2) (x + 2))$

Step 6.2

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

$2 x ((x - 2) (x - 2) (x + 2))$

Step 6.3

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

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

Step 6.4

Add $1$ and $1$ .

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