Algebra Examples

$x^{4} + 3 x^{3} - 4 x^{2} - 16 x^{2} - 48 x + 64$

Step 1

Regroup terms.

$x^{4} + 3 x^{3} - 4 x^{2} - 16 x^{2} - 48 x + 64$

Step 2

Factor $x^{2}$ out of $x^{4} + 3 x^{3} - 4 x^{2}$ .

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Step 2.1

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

$x^{2} x^{2} + 3 x^{3} - 4 x^{2} - 16 x^{2} - 48 x + 64$

Step 2.2

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

$x^{2} x^{2} + x^{2} (3 x) - 4 x^{2} - 16 x^{2} - 48 x + 64$

Step 2.3

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

$x^{2} x^{2} + x^{2} (3 x) + x^{2} \cdot - 4 - 16 x^{2} - 48 x + 64$

Step 2.4

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (3 x)$ .

$x^{2} (x^{2} + 3 x) + x^{2} \cdot - 4 - 16 x^{2} - 48 x + 64$

Step 2.5

Factor $x^{2}$ out of $x^{2} (x^{2} + 3 x) + x^{2} \cdot - 4$ .

$x^{2} (x^{2} + 3 x - 4) - 16 x^{2} - 48 x + 64$

Step 3

Factor.

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Step 3.1

Factor $x^{2} + 3 x - 4$ using the AC method.

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Step 3.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 3.1.2

Write the factored form using these integers.

$x^{2} ((x - 1) (x + 4)) - 16 x^{2} - 48 x + 64$

Step 3.2

Remove unnecessary parentheses.

$x^{2} (x - 1) (x + 4) - 16 x^{2} - 48 x + 64$

Step 4

Factor $16$ out of $- 16 x^{2} - 48 x + 64$ .

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Step 4.1

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

$x^{2} (x - 1) (x + 4) + 16 (- x^{2}) - 48 x + 64$

Step 4.2

Factor $16$ out of $- 48 x$ .

$x^{2} (x - 1) (x + 4) + 16 (- x^{2}) + 16 (- 3 x) + 64$

Step 4.3

Factor $16$ out of $64$ .

$x^{2} (x - 1) (x + 4) + 16 (- x^{2}) + 16 (- 3 x) + 16 (4)$

Step 4.4

Factor $16$ out of $16 (- x^{2}) + 16 (- 3 x)$ .

$x^{2} (x - 1) (x + 4) + 16 (- x^{2} - 3 x) + 16 (4)$

Step 4.5

Factor $16$ out of $16 (- x^{2} - 3 x) + 16 (4)$ .

$x^{2} (x - 1) (x + 4) + 16 (- x^{2} - 3 x + 4)$

Step 5

Factor.

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Step 5.1

Factor by grouping.

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Step 5.1.1

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 $a \cdot c = - 1 \cdot 4 = - 4$ and whose sum is $b = - 3$ .

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Step 5.1.1.1

Factor $- 3$ out of $- 3 x$ .

$x^{2} (x - 1) (x + 4) + 16 (- x^{2} - 3 (x) + 4)$

Step 5.1.1.2

Rewrite $- 3$ as $1$ plus $- 4$

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

Step 5.1.1.3

Apply the distributive property.

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

Step 5.1.1.4

Multiply $x$ by $1$ .

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

Step 5.1.2

Factor out the greatest common factor from each group.

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Step 5.1.2.1

Group the first two terms and the last two terms.

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

Step 5.1.2.2

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

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

Step 5.1.3

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

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

Step 5.2

Remove unnecessary parentheses.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Factor $x + 4$ out of $16 (- x + 1) (x + 4)$ .

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

Step 6.3

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

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

Step 7

Apply the distributive property.

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

Step 8

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 8.1

Multiply $x^{2}$ by $x$ .

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Step 8.1.1

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

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

Step 8.1.2

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

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

Step 8.2

Add $2$ and $1$ .

$(x + 4) (x^{3} + x^{2} \cdot - 1 + 16 (- x + 1))$

Step 9

Move $- 1$ to the left of $x^{2}$ .

$(x + 4) (x^{3} - 1 x^{2} + 16 (- x + 1))$

Step 10

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$(x + 4) (x^{3} - x^{2} + 16 (- x + 1))$

Step 11

Apply the distributive property.

$(x + 4) (x^{3} - x^{2} + 16 (- x) + 16 \cdot 1)$

Step 12

Multiply $- 1$ by $16$ .

$(x + 4) (x^{3} - x^{2} - 16 x + 16 \cdot 1)$

Step 13

Multiply $16$ by $1$ .

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

Step 14

Factor.

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Step 14.1

Rewrite $x^{3} - x^{2} - 16 x + 16$ in a factored form.

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Step 14.1.1

Factor out the greatest common factor from each group.

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Step 14.1.1.1

Group the first two terms and the last two terms.

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

Step 14.1.1.2

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

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

Step 14.1.2

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

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

Step 14.1.3

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

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

Step 14.1.4

Factor.

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Step 14.1.4.1

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 $b = 4$ .

$(x + 4) ((x - 1) ((x + 4) (x - 4)))$

Step 14.1.4.2

Remove unnecessary parentheses.

$(x + 4) ((x - 1) (x + 4) (x - 4))$

Step 14.2

Remove unnecessary parentheses.

$(x + 4) (x - 1) (x + 4) (x - 4)$

Step 15

Combine exponents.

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Step 15.1

Raise $x + 4$ to the power of $1$ .

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

Step 15.2

Raise $x + 4$ to the power of $1$ .

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

Step 15.3

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

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

Step 15.4

Add $1$ and $1$ .

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