Trigonometry Examples

$f (x) = 9 x^{4} + 10 x^{3} + 19 x^{2} + 20 x + 2$

Step 1

Regroup terms.

$f (x) = 10 x^{3} + 20 x + 9 x^{4} + 19 x^{2} + 2$

Step 2

Factor $10 x$ out of $10 x^{3} + 20 x$ .

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Step 2.1

Factor $10 x$ out of $10 x^{3}$ .

$f (x) = 10 x (x^{2}) + 20 x + 9 x^{4} + 19 x^{2} + 2$

Step 2.2

Factor $10 x$ out of $20 x$ .

$f (x) = 10 x (x^{2}) + 10 x (2) + 9 x^{4} + 19 x^{2} + 2$

Step 2.3

Factor $10 x$ out of $10 x (x^{2}) + 10 x (2)$ .

$f (x) = 10 x (x^{2} + 2) + 9 x^{4} + 19 x^{2} + 2$

Step 3

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

$f (x) = 10 x (x^{2} + 2) + 9 {(x^{2})}^{2} + 19 x^{2} + 2$

Step 4

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

$f (x) = 10 x (x^{2} + 2) + 9 u^{2} + 19 u + 2$

Step 5

Factor by grouping.

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Step 5.1

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 $a \cdot c = 9 \cdot 2 = 18$ and whose sum is $b = 19$ .

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Step 5.1.1

Factor $19$ out of $19 u$ .

$f (x) = 10 x (x^{2} + 2) + 9 u^{2} + 19 (u) + 2$

Step 5.1.2

Rewrite $19$ as $1$ plus $18$

$f (x) = 10 x (x^{2} + 2) + 9 u^{2} + (1 + 18) u + 2$

Step 5.1.3

Apply the distributive property.

$f (x) = 10 x (x^{2} + 2) + 9 u^{2} + 1 u + 18 u + 2$

Step 5.1.4

Multiply $u$ by $1$ .

$f (x) = 10 x (x^{2} + 2) + 9 u^{2} + u + 18 u + 2$

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

$f (x) = 10 x (x^{2} + 2) + 9 u^{2} + u + 18 u + 2$

Step 5.2.2

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

$f (x) = 10 x (x^{2} + 2) + u (9 u + 1) + 2 (9 u + 1)$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $9 u + 1$ .

$f (x) = 10 x (x^{2} + 2) + (9 u + 1) (u + 2)$

Step 6

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

$f (x) = 10 x (x^{2} + 2) + (9 x^{2} + 1) (x^{2} + 2)$

Step 7

Factor $x^{2} + 2$ out of $10 x (x^{2} + 2) + (9 x^{2} + 1) (x^{2} + 2)$ .

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Step 7.1

Factor $x^{2} + 2$ out of $10 x (x^{2} + 2)$ .

$f (x) = (x^{2} + 2) (10 x) + (9 x^{2} + 1) (x^{2} + 2)$

Step 7.2

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

$f (x) = (x^{2} + 2) (10 x) + (x^{2} + 2) (9 x^{2} + 1)$

Step 7.3

Factor $x^{2} + 2$ out of $(x^{2} + 2) (10 x) + (x^{2} + 2) (9 x^{2} + 1)$ .

$f (x) = (x^{2} + 2) (10 x + 9 x^{2} + 1)$

Step 8

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

$f (x) = (x^{2} + 2) (10 u + 9 u^{2} + 1)$

Step 9

Factor by grouping.

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Step 9.1

Reorder terms.

$f (x) = (x^{2} + 2) (9 u^{2} + 10 u + 1)$

Step 9.2

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 $a \cdot c = 9 \cdot 1 = 9$ and whose sum is $b = 10$ .

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Step 9.2.1

Factor $10$ out of $10 u$ .

$f (x) = (x^{2} + 2) (9 u^{2} + 10 (u) + 1)$

Step 9.2.2

Rewrite $10$ as $1$ plus $9$

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

Step 9.2.3

Apply the distributive property.

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

Step 9.2.4

Multiply $u$ by $1$ .

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

Step 9.3

Factor out the greatest common factor from each group.

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Step 9.3.1

Group the first two terms and the last two terms.

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

Step 9.3.2

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

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

Step 9.4

Factor the polynomial by factoring out the greatest common factor, $9 u + 1$ .

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

Step 10

Factor.

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Step 10.1

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

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

Step 10.2

Remove unnecessary parentheses.

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