Algebra Examples

$x^{4} + 7 x^{3} - 18 x^{2} - x^{2} - 7 x + 18$

Step 1

Regroup terms.

$x^{4} + 7 x^{3} - 18 x^{2} - x^{2} - 7 x + 18$

Step 2

Factor $x^{2}$ out of $x^{4} + 7 x^{3} - 18 x^{2}$ .

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Step 2.1

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

$x^{2} x^{2} + 7 x^{3} - 18 x^{2} - x^{2} - 7 x + 18$

Step 2.2

Factor $x^{2}$ out of $7 x^{3}$ .

$x^{2} x^{2} + x^{2} (7 x) - 18 x^{2} - x^{2} - 7 x + 18$

Step 2.3

Factor $x^{2}$ out of $- 18 x^{2}$ .

$x^{2} x^{2} + x^{2} (7 x) + x^{2} \cdot - 18 - x^{2} - 7 x + 18$

Step 2.4

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (7 x)$ .

$x^{2} (x^{2} + 7 x) + x^{2} \cdot - 18 - x^{2} - 7 x + 18$

Step 2.5

Factor $x^{2}$ out of $x^{2} (x^{2} + 7 x) + x^{2} \cdot - 18$ .

$x^{2} (x^{2} + 7 x - 18) - x^{2} - 7 x + 18$

Step 3

Factor.

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Step 3.1

Factor $x^{2} + 7 x - 18$ 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 $- 18$ and whose sum is $7$ .

$- 2, 9$

Step 3.1.2

Write the factored form using these integers.

$x^{2} ((x - 2) (x + 9)) - x^{2} - 7 x + 18$

Step 3.2

Remove unnecessary parentheses.

$x^{2} (x - 2) (x + 9) - x^{2} - 7 x + 18$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = - 1 \cdot 18 = - 18$ and whose sum is $b = - 7$ .

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Step 4.1.1

Factor $- 7$ out of $- 7 x$ .

$x^{2} (x - 2) (x + 9) - x^{2} - 7 (x) + 18$

Step 4.1.2

Rewrite $- 7$ as $2$ plus $- 9$

$x^{2} (x - 2) (x + 9) - x^{2} + (2 - 9) x + 18$

Step 4.1.3

Apply the distributive property.

$x^{2} (x - 2) (x + 9) - x^{2} + 2 x - 9 x + 18$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$x^{2} (x - 2) (x + 9) + (- x^{2} + 2 x) - 9 x + 18$

Step 4.2.2

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

$x^{2} (x - 2) (x + 9) + x (- x + 2) + 9 (- x + 2)$

Step 4.3

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

$x^{2} (x - 2) (x + 9) + (- x + 2) (x + 9)$

Step 5

Factor $x + 9$ out of $x^{2} (x - 2) (x + 9) + (- x + 2) (x + 9)$ .

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Step 5.1

Factor $x + 9$ out of $x^{2} (x - 2) (x + 9)$ .

$(x + 9) (x^{2} (x - 2)) + (- x + 2) (x + 9)$

Step 5.2

Factor $x + 9$ out of $(- x + 2) (x + 9)$ .

$(x + 9) (x^{2} (x - 2)) + (x + 9) (- x + 2)$

Step 5.3

Factor $x + 9$ out of $(x + 9) (x^{2} (x - 2)) + (x + 9) (- x + 2)$ .

$(x + 9) (x^{2} (x - 2) - x + 2)$

Step 6

Apply the distributive property.

$(x + 9) (x^{2} x + x^{2} \cdot - 2 - x + 2)$

Step 7

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

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Step 7.1

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

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Step 7.1.1

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

$(x + 9) (x^{2} x^{1} + x^{2} \cdot - 2 - x + 2)$

Step 7.1.2

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

$(x + 9) (x^{2 + 1} + x^{2} \cdot - 2 - x + 2)$

Step 7.2

Add $2$ and $1$ .

$(x + 9) (x^{3} + x^{2} \cdot - 2 - x + 2)$

Step 8

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

$(x + 9) (x^{3} - 2 x^{2} - x + 2)$

Step 9

Factor.

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Step 9.1

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

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Step 9.1.1

Factor out the greatest common factor from each group.

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Step 9.1.1.1

Group the first two terms and the last two terms.

$(x + 9) ((x^{3} - 2 x^{2}) - x + 2)$

Step 9.1.1.2

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

$(x + 9) (x^{2} (x - 2) - (x - 2))$

Step 9.1.2

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

$(x + 9) ((x - 2) (x^{2} - 1))$

Step 9.1.3

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

$(x + 9) ((x - 2) (x^{2} - 1^{2}))$

Step 9.1.4

Factor.

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Step 9.1.4.1

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 $b = 1$ .

$(x + 9) ((x - 2) ((x + 1) (x - 1)))$

Step 9.1.4.2

Remove unnecessary parentheses.

$(x + 9) ((x - 2) (x + 1) (x - 1))$

Step 9.2

Remove unnecessary parentheses.

$(x + 9) (x - 2) (x + 1) (x - 1)$