Write as a Set of Linear Factors f(x)=x^4-2x^3-17x^2+18x+72

$f (x) = x^{4} - 2 x^{3} - 17 x^{2} + 18 x + 72$

Step 1

Regroup terms.

$f (x) = - 2 x^{3} + 18 x + x^{4} - 17 x^{2} + 72$

Step 2

Factor $- 2 x$ out of $- 2 x^{3} + 18 x$ .

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

$f (x) = - 2 x \cdot x^{2} + 18 x + x^{4} - 17 x^{2} + 72$

Factor $- 2 x$ out of $18 x$ .

$f (x) = - 2 x \cdot x^{2} - 2 x \cdot - 9 + x^{4} - 17 x^{2} + 72$

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

$f (x) = - 2 x (x^{2} - 9) + x^{4} - 17 x^{2} + 72$

Step 3

Rewrite $9$ as $3^{2}$ .

$f (x) = - 2 x (x^{2} - 3^{2}) + x^{4} - 17 x^{2} + 72$

Step 4

Factor.

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 $i = 3$ .

$f (x) = - 2 x ((x + 3) (x - 3)) + x^{4} - 17 x^{2} + 72$

Remove unnecessary parentheses.

$f (x) = - 2 x (x + 3) (x - 3) + x^{4} - 17 x^{2} + 72$

Step 5

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

$f (x) = - 2 x (x + 3) (x - 3) + {(x^{2})}^{2} - 17 x^{2} + 72$

Step 6

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

$f (x) = - 2 x (x + 3) (x - 3) + u^{2} - 17 u + 72$

Step 7

Factor $u^{2} - 17 u + 72$ using the AC method.

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 $72$ and whose sum is $- 17$ .

$f (x) = - 9, - 8$

Write the factored form using these integers.

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

Step 8

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

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

Step 9

Rewrite $9$ as $3^{2}$ .

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

Step 10

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 $i = 3$ .

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

Step 11

Factor $(x + 3) (x - 3)$ out of $- 2 x (x + 3) (x - 3) + (x + 3) (x - 3) (x^{2} - 8)$ .

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

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

Factor $(x + 3) (x - 3)$ out of $(x + 3) (x - 3) (- 2 x) + (x + 3) (x - 3) (x^{2} - 8)$ .

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

Step 12

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

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

Step 13

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

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 $- 8$ and whose sum is $- 2$ .

$f (x) = - 4, 2$

Write the factored form using these integers.

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

Step 14

Factor.

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

$f (x) = (x + 3) (x - 3) ((x - 4) (x + 2))$

Remove unnecessary parentheses.

$f (x) = (x + 3) (x - 3) (x - 4) (x + 2)$

Write as a Set of Linear Factors f(x)=x^4-2x^3-17x^2+18x+72

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