Algebra Examples

$f (x) = x^{4} - 4 x^{3} - 21 x^{2} + 100 x - 100$

Step 1

Regroup terms.

$f (x) = - 4 x^{3} + 100 x + x^{4} - 21 x^{2} - 100$

Step 2

Factor $- 4 x$ out of $- 4 x^{3} + 100 x$ .

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Step 2.1

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

$f (x) = - 4 x \cdot x^{2} + 100 x + x^{4} - 21 x^{2} - 100$

Step 2.2

Factor $- 4 x$ out of $100 x$ .

$f (x) = - 4 x \cdot x^{2} - 4 x \cdot - 25 + x^{4} - 21 x^{2} - 100$

Step 2.3

Factor $- 4 x$ out of $- 4 x (x^{2}) - 4 x (- 25)$ .

$f (x) = - 4 x (x^{2} - 25) + x^{4} - 21 x^{2} - 100$

Step 3

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

$f (x) = - 4 x (x^{2} - 5^{2}) + x^{4} - 21 x^{2} - 100$

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 $b = 5$ .

$f (x) = - 4 x ((x + 5) (x - 5)) + x^{4} - 21 x^{2} - 100$

Step 4.2

Remove unnecessary parentheses.

$f (x) = - 4 x (x + 5) (x - 5) + x^{4} - 21 x^{2} - 100$

Step 5

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

$f (x) = - 4 x (x + 5) (x - 5) + {(x^{2})}^{2} - 21 x^{2} - 100$

Step 6

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

$f (x) = - 4 x (x + 5) (x - 5) + u^{2} - 21 u - 100$

Step 7

Factor $u^{2} - 21 u - 100$ 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 $- 100$ and whose sum is $- 21$ .

$- 25, 4$

Step 7.2

Write the factored form using these integers.

$f (x) = - 4 x (x + 5) (x - 5) + (u - 25) (u + 4)$

Step 8

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

$f (x) = - 4 x (x + 5) (x - 5) + (x^{2} - 25) (x^{2} + 4)$

Step 9

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

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

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 $b = 5$ .

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

Step 11

Factor $(x + 5) (x - 5)$ out of $- 4 x (x + 5) (x - 5) + (x + 5) (x - 5) (x^{2} + 4)$ .

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Step 11.1

Factor $(x + 5) (x - 5)$ out of $- 4 x (x + 5) (x - 5)$ .

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

Step 11.2

Factor $(x + 5) (x - 5)$ out of $(x + 5) (x - 5) (- 4 x) + (x + 5) (x - 5) (x^{2} + 4)$ .

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

Step 12

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

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

Step 13

Factor using the perfect square rule.

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Step 13.1

Rearrange terms.

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

Step 13.2

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

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

Step 13.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot u \cdot 2$

Step 13.4

Rewrite the polynomial.

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

Step 13.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 2$ .

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

Step 14

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

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