Trigonometry Examples

$f (x) = 9 x^{4} - 9 x^{3} - 58 x^{2} + 4 x + 24$

Step 1

Regroup terms.

$f (x) = - 9 x^{3} + 4 x + 9 x^{4} - 58 x^{2} + 24$

Step 2

Factor $- x$ out of $- 9 x^{3} + 4 x$ .

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Step 2.1

Factor $- x$ out of $- 9 x^{3}$ .

$f (x) = - x (9 x^{2}) + 4 x + 9 x^{4} - 58 x^{2} + 24$

Step 2.2

Factor $- x$ out of $4 x$ .

$f (x) = - x (9 x^{2}) - x \cdot - 4 + 9 x^{4} - 58 x^{2} + 24$

Step 2.3

Factor $- x$ out of $- x (9 x^{2}) - x (- 4)$ .

$f (x) = - x (9 x^{2} - 4) + 9 x^{4} - 58 x^{2} + 24$

Step 3

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$f (x) = - x ({(3 x)}^{2} - 4) + 9 x^{4} - 58 x^{2} + 24$

Step 4

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

$f (x) = - x ({(3 x)}^{2} - 2^{2}) + 9 x^{4} - 58 x^{2} + 24$

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 $a = 3 x$ and $b = 2$ .

$f (x) = - x ((3 x + 2) (3 x - 2)) + 9 x^{4} - 58 x^{2} + 24$

Step 5.2

Remove unnecessary parentheses.

$f (x) = - x (3 x + 2) (3 x - 2) + 9 x^{4} - 58 x^{2} + 24$

Step 6

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

$f (x) = - x (3 x + 2) (3 x - 2) + 9 {(x^{2})}^{2} - 58 x^{2} + 24$

Step 7

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

$f (x) = - x (3 x + 2) (3 x - 2) + 9 u^{2} - 58 u + 24$

Step 8

Factor by grouping.

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Step 8.1

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 $a \cdot c = 9 \cdot 24 = 216$ and whose sum is $b = - 58$ .

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Step 8.1.1

Factor $- 58$ out of $- 58 u$ .

$f (x) = - x (3 x + 2) (3 x - 2) + 9 u^{2} - 58 u + 24$

Step 8.1.2

Rewrite $- 58$ as $- 4$ plus $- 54$

$f (x) = - x (3 x + 2) (3 x - 2) + 9 u^{2} + (- 4 - 54) u + 24$

Step 8.1.3

Apply the distributive property.

$f (x) = - x (3 x + 2) (3 x - 2) + 9 u^{2} - 4 u - 54 u + 24$

Step 8.2

Factor out the greatest common factor from each group.

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Step 8.2.1

Group the first two terms and the last two terms.

$f (x) = - x (3 x + 2) (3 x - 2) + 9 u^{2} - 4 u - 54 u + 24$

Step 8.2.2

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

$f (x) = - x (3 x + 2) (3 x - 2) + u (9 u - 4) - 6 (9 u - 4)$

Step 8.3

Factor the polynomial by factoring out the greatest common factor, $9 u - 4$ .

$f (x) = - x (3 x + 2) (3 x - 2) + (9 u - 4) (u - 6)$

Step 9

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

$f (x) = - x (3 x + 2) (3 x - 2) + (9 x^{2} - 4) (x^{2} - 6)$

Step 10

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$f (x) = - x (3 x + 2) (3 x - 2) + ({(3 x)}^{2} - 4) (x^{2} - 6)$

Step 11

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

$f (x) = - x (3 x + 2) (3 x - 2) + ({(3 x)}^{2} - 2^{2}) (x^{2} - 6)$

Step 12

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 2$ .

$f (x) = - x (3 x + 2) (3 x - 2) + (3 x + 2) (3 x - 2) (x^{2} - 6)$

Step 13

Factor $(3 x + 2) (3 x - 2)$ out of $- x (3 x + 2) (3 x - 2) + (3 x + 2) (3 x - 2) (x^{2} - 6)$ .

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Step 13.1

Factor $(3 x + 2) (3 x - 2)$ out of $- x (3 x + 2) (3 x - 2)$ .

$f (x) = (3 x + 2) (3 x - 2) (- x) + (3 x + 2) (3 x - 2) (x^{2} - 6)$

Step 13.2

Factor $(3 x + 2) (3 x - 2)$ out of $(3 x + 2) (3 x - 2) (- x) + (3 x + 2) (3 x - 2) (x^{2} - 6)$ .

$f (x) = (3 x + 2) (3 x - 2) (- x + x^{2} - 6)$

Step 14

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$f (x) = (3 x + 2) (3 x - 2) (- u + u^{2} - 6)$

Step 15

Factor $- u + u^{2} - 6$ using the AC method.

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Step 15.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 15.2

Write the factored form using these integers.

$f (x) = (3 x + 2) (3 x - 2) ((u - 3) (u + 2))$

Step 16

Factor.

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Step 16.1

Replace all occurrences of $u$ with $x$ .

$f (x) = (3 x + 2) (3 x - 2) ((x - 3) (x + 2))$

Step 16.2

Remove unnecessary parentheses.

$f (x) = (3 x + 2) (3 x - 2) (x - 3) (x + 2)$