Trigonometry Examples

$f (x) = x^{4} - 2 x^{3} - 28 x^{2} + 50 x + 75$

Step 1

Regroup terms.

$f (x) = - 2 x^{3} + 50 x + x^{4} - 28 x^{2} + 75$

Step 2

Factor $- 2 x$ out of $- 2 x^{3} + 50 x$ .

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Step 2.1

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

$f (x) = - 2 x \cdot x^{2} + 50 x + x^{4} - 28 x^{2} + 75$

Step 2.2

Factor $- 2 x$ out of $50 x$ .

$f (x) = - 2 x \cdot x^{2} - 2 x \cdot - 25 + x^{4} - 28 x^{2} + 75$

Step 2.3

Factor $- 2 x$ out of $- 2 x (x^{2}) - 2 x (- 25)$ .

$f (x) = - 2 x (x^{2} - 25) + x^{4} - 28 x^{2} + 75$

Step 3

Rewrite $25$ as $5^{2}$ .

$f (x) = - 2 x (x^{2} - 5^{2}) + x^{4} - 28 x^{2} + 75$

Step 4

Factor.

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Step 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 $b = 5$ .

$f (x) = - 2 x ((x + 5) (x - 5)) + x^{4} - 28 x^{2} + 75$

Step 4.2

Remove unnecessary parentheses.

$f (x) = - 2 x (x + 5) (x - 5) + x^{4} - 28 x^{2} + 75$

Step 5

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

$f (x) = - 2 x (x + 5) (x - 5) + {(x^{2})}^{2} - 28 x^{2} + 75$

Step 6

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

$f (x) = - 2 x (x + 5) (x - 5) + u^{2} - 28 u + 75$

Step 7

Factor $u^{2} - 28 u + 75$ using the AC method.

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Step 7.1

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 $75$ and whose sum is $- 28$ .

$- 25, - 3$

Step 7.2

Write the factored form using these integers.

$f (x) = - 2 x (x + 5) (x - 5) + (u - 25) (u - 3)$

Step 8

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

$f (x) = - 2 x (x + 5) (x - 5) + (x^{2} - 25) (x^{2} - 3)$

Step 9

Rewrite $25$ as $5^{2}$ .

$f (x) = - 2 x (x + 5) (x - 5) + (x^{2} - 5^{2}) (x^{2} - 3)$

Step 10

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 $b = 5$ .

$f (x) = - 2 x (x + 5) (x - 5) + (x + 5) (x - 5) (x^{2} - 3)$

Step 11

Factor $(x + 5) (x - 5)$ out of $- 2 x (x + 5) (x - 5) + (x + 5) (x - 5) (x^{2} - 3)$ .

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Step 11.1

Factor $(x + 5) (x - 5)$ out of $- 2 x (x + 5) (x - 5)$ .

$f (x) = (x + 5) (x - 5) (- 2 x) + (x + 5) (x - 5) (x^{2} - 3)$

Step 11.2

Factor $(x + 5) (x - 5)$ out of $(x + 5) (x - 5) (- 2 x) + (x + 5) (x - 5) (x^{2} - 3)$ .

$f (x) = (x + 5) (x - 5) (- 2 x + x^{2} - 3)$

Step 12

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

$f (x) = (x + 5) (x - 5) (- 2 u + u^{2} - 3)$

Step 13

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

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Step 13.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 13.2

Write the factored form using these integers.

$f (x) = (x + 5) (x - 5) ((u - 3) (u + 1))$

Step 14

Factor.

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Step 14.1

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

$f (x) = (x + 5) (x - 5) ((x - 3) (x + 1))$

Step 14.2

Remove unnecessary parentheses.

$f (x) = (x + 5) (x - 5) (x - 3) (x + 1)$