Algebra Examples

$f (x) = 2 x^{4} + 3 x^{3} + 16 x^{2} + 27 x - 18$

Step 1

Regroup terms.

$f (x) = 3 x^{3} + 27 x + 2 x^{4} + 16 x^{2} - 18$

Step 2

Factor $3 x$ out of $3 x^{3} + 27 x$ .

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Step 2.1

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

$f (x) = 3 x (x^{2}) + 27 x + 2 x^{4} + 16 x^{2} - 18$

Step 2.2

Factor $3 x$ out of $27 x$ .

$f (x) = 3 x (x^{2}) + 3 x (9) + 2 x^{4} + 16 x^{2} - 18$

Step 2.3

Factor $3 x$ out of $3 x (x^{2}) + 3 x (9)$ .

$f (x) = 3 x (x^{2} + 9) + 2 x^{4} + 16 x^{2} - 18$

Step 3

Factor $2$ out of $2 x^{4} + 16 x^{2} - 18$ .

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Step 3.1

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

$f (x) = 3 x (x^{2} + 9) + 2 (x^{4}) + 16 x^{2} - 18$

Step 3.2

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

$f (x) = 3 x (x^{2} + 9) + 2 (x^{4}) + 2 (8 x^{2}) - 18$

Step 3.3

Factor $2$ out of $- 18$ .

$f (x) = 3 x (x^{2} + 9) + 2 x^{4} + 2 (8 x^{2}) + 2 \cdot - 9$

Step 3.4

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

$f (x) = 3 x (x^{2} + 9) + 2 (x^{4} + 8 x^{2}) + 2 \cdot - 9$

Step 3.5

Factor $2$ out of $2 (x^{4} + 8 x^{2}) + 2 \cdot - 9$ .

$f (x) = 3 x (x^{2} + 9) + 2 (x^{4} + 8 x^{2} - 9)$

Step 4

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

$f (x) = 3 x (x^{2} + 9) + 2 ({(x^{2})}^{2} + 8 x^{2} - 9)$

Step 5

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

$f (x) = 3 x (x^{2} + 9) + 2 (u^{2} + 8 u - 9)$

Step 6

Factor $u^{2} + 8 u - 9$ using the AC method.

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Step 6.1

Consider the form $x^{2} + i x + c$ . Find a pair of integers whose product is $c$ and whose sum is $i$ . In this case, whose product is $- 9$ and whose sum is $8$ .

$f (x) = - 1, 9$

Step 6.2

Write the factored form using these integers.

$f (x) = 3 x (x^{2} + 9) + 2 ((u - 1) (u + 9))$

Step 7

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

$f (x) = 3 x (x^{2} + 9) + 2 ((x^{2} - 1) (x^{2} + 9))$

Step 8

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

$f (x) = 3 x (x^{2} + 9) + 2 ((x^{2} - 1^{2}) (x^{2} + 9))$

Step 9

Factor.

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Step 9.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - i^{2} = (a + i) (a - i)$ where $a = x$ and $i = 1$ .

$f (x) = 3 x (x^{2} + 9) + 2 ((x + 1) (x - 1) (x^{2} + 9))$

Step 9.2

Remove unnecessary parentheses.

$f (x) = 3 x (x^{2} + 9) + 2 (x + 1) (x - 1) (x^{2} + 9)$

Step 10

Factor $x^{2} + 9$ out of $3 x (x^{2} + 9) + 2 (x + 1) (x - 1) (x^{2} + 9)$ .

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Step 10.1

Factor $x^{2} + 9$ out of $3 x (x^{2} + 9)$ .

$f (x) = (x^{2} + 9) (3 x) + 2 (x + 1) (x - 1) (x^{2} + 9)$

Step 10.2

Factor $x^{2} + 9$ out of $2 (x + 1) (x - 1) (x^{2} + 9)$ .

$f (x) = (x^{2} + 9) (3 x) + (x^{2} + 9) (2 (x + 1) (x - 1))$

Step 10.3

Factor $x^{2} + 9$ out of $(x^{2} + 9) (3 x) + (x^{2} + 9) (2 (x + 1) (x - 1))$ .

$f (x) = (x^{2} + 9) (3 x + 2 (x + 1) (x - 1))$

Step 11

Apply the distributive property.

$f (x) = (x^{2} + 9) (3 x + (2 x + 2 \cdot 1) (x - 1))$

Step 12

Multiply $2$ by $1$ .

$f (x) = (x^{2} + 9) (3 x + (2 x + 2) (x - 1))$

Step 13

Expand $(2 x + 2) (x - 1)$ using the FOIL Method.

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Step 13.1

Apply the distributive property.

$f (x) = (x^{2} + 9) (3 x + 2 x (x - 1) + 2 (x - 1))$

Step 13.2

Apply the distributive property.

$f (x) = (x^{2} + 9) (3 x + 2 x \cdot x + 2 x \cdot - 1 + 2 (x - 1))$

Step 13.3

Apply the distributive property.

$f (x) = (x^{2} + 9) (3 x + 2 x \cdot x + 2 x \cdot - 1 + 2 x + 2 \cdot - 1)$

Step 14

Simplify and combine like terms.

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Step 14.1

Simplify each term.

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Step 14.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 14.1.1.1

Move $x$ .

$f (x) = (x^{2} + 9) (3 x + 2 (x \cdot x) + 2 x \cdot - 1 + 2 x + 2 \cdot - 1)$

Step 14.1.1.2

Multiply $x$ by $x$ .

$f (x) = (x^{2} + 9) (3 x + 2 x^{2} + 2 x \cdot - 1 + 2 x + 2 \cdot - 1)$

Step 14.1.2

Multiply $- 1$ by $2$ .

$f (x) = (x^{2} + 9) (3 x + 2 x^{2} - 2 x + 2 x + 2 \cdot - 1)$

Step 14.1.3

Multiply $2$ by $- 1$ .

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

Step 14.2

Add $- 2 x$ and $2 x$ .

$f (x) = (x^{2} + 9) (3 x + 2 x^{2} + 0 - 2)$

Step 14.3

Add $2 x^{2}$ and $0$ .

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

Step 15

Reorder terms.

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

Step 16

Factor.

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Step 16.1

Factor by grouping.

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Step 16.1.1

For a polynomial of the form $a x^{2} + i x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $i = 3$ .

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Step 16.1.1.1

Factor $3$ out of $3 x$ .

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

Step 16.1.1.2

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

$f (x) = (x^{2} + 9) (2 x^{2} + (- 1 + 4) x - 2)$

Step 16.1.1.3

Apply the distributive property.

$f (x) = (x^{2} + 9) (2 x^{2} - 1 x + 4 x - 2)$

Step 16.1.2

Factor out the greatest common factor from each group.

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Step 16.1.2.1

Group the first two terms and the last two terms.

$f (x) = (x^{2} + 9) ((2 x^{2} - 1 x) + 4 x - 2)$

Step 16.1.2.2

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

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

Step 16.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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

Step 16.2

Remove unnecessary parentheses.

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