Write as a Set of Linear Factors f(x)=x^4+x^3+3x^2+5x-10

$f (x) = x^{4} + x^{3} + 3 x^{2} + 5 x - 10$

Step 1

Regroup terms.

$f (x) = x^{3} + 5 x + x^{4} + 3 x^{2} - 10$

Step 2

Factor $x$ out of $x^{3} + 5 x$ .

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

$f (x) = x \cdot x^{2} + 5 x + x^{4} + 3 x^{2} - 10$

Factor $x$ out of $5 x$ .

$f (x) = x \cdot x^{2} + x \cdot 5 + x^{4} + 3 x^{2} - 10$

Factor $x$ out of $x \cdot x^{2} + x \cdot 5$ .

$f (x) = x (x^{2} + 5) + x^{4} + 3 x^{2} - 10$

Step 3

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

$f (x) = x (x^{2} + 5) + {(x^{2})}^{2} + 3 x^{2} - 10$

Step 4

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

$f (x) = x (x^{2} + 5) + u^{2} + 3 u - 10$

Step 5

Factor $u^{2} + 3 u - 10$ 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 $- 10$ and whose sum is $3$ .

$f (x) = - 2, 5$

Write the factored form using these integers.

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

Step 6

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

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

Step 7

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

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

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

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

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

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

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

Step 8

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

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

Step 9

Factor $u + u^{2} - 2$ 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 $- 2$ and whose sum is $1$ .

$f (x) = - 1, 2$

Write the factored form using these integers.

$f (x) = (x^{2} + 5) ((u - 1) (u + 2))$

Step 10

Factor.

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

$f (x) = (x^{2} + 5) ((x - 1) (x + 2))$

Remove unnecessary parentheses.

$f (x) = (x^{2} + 5) (x - 1) (x + 2)$

Write as a Set of Linear Factors f(x)=x^4+x^3+3x^2+5x-10

Related Questions

Support LCMGCF.com by disabling your adblocker.