Precalculus Examples

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

Step 1

Factor out the greatest common factor from each group.

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Step 1.1

Group the first two terms and the last two terms.

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

Step 1.2

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

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

Step 2

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

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

Step 3

Rewrite $8$ as $2^{3}$ .

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

Step 4

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 $a = x$ and $b = 2$ .

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

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Move $2$ to the left of $x$ .

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

Step 5.1.2

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

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

Step 5.2

Remove unnecessary parentheses.

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

Step 6

Combine exponents.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Add $1$ and $1$ .

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