Write as a Set of Linear Factors f(x)=7x^4-6x^3+174x^2-150x-25

$f (x) = 7 x^{4} - 6 x^{3} + 174 x^{2} - 150 x - 25$

Step 1

Regroup terms.

$f (x) = - 6 x^{3} - 150 x + 7 x^{4} + 174 x^{2} - 25$

Step 2

Factor $- 6 x$ out of $- 6 x^{3} - 150 x$ .

Factor $- 6 x$ out of $- 6 x^{3}$ .

$f (x) = - 6 x \cdot x^{2} - 150 x + 7 x^{4} + 174 x^{2} - 25$

Factor $- 6 x$ out of $- 150 x$ .

$f (x) = - 6 x \cdot x^{2} - 6 x \cdot 25 + 7 x^{4} + 174 x^{2} - 25$

Factor $- 6 x$ out of $- 6 x (x^{2}) - 6 x (25)$ .

$f (x) = - 6 x (x^{2} + 25) + 7 x^{4} + 174 x^{2} - 25$

Step 3

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

$f (x) = - 6 x (x^{2} + 25) + 7 {(x^{2})}^{2} + 174 x^{2} - 25$

Step 4

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

$f (x) = - 6 x (x^{2} + 25) + 7 u^{2} + 174 u - 25$

Step 5

Factor by grouping.

For a polynomial of the form $a x^{2} + i x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 7 \cdot - 25 = - 175$ and whose sum is $i = 174$ .

Factor $174$ out of $174 u$ .

$f (x) = - 6 x (x^{2} + 25) + 7 u^{2} + 174 (u) - 25$

Rewrite $174$ as $- 1$ plus $175$

$f (x) = - 6 x (x^{2} + 25) + 7 u^{2} + (- 1 + 175) u - 25$

Apply the distributive property.

$f (x) = - 6 x (x^{2} + 25) + 7 u^{2} - 1 u + 175 u - 25$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = - 6 x (x^{2} + 25) + 7 u^{2} - 1 u + 175 u - 25$

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

$f (x) = - 6 x (x^{2} + 25) + u (7 u - 1) + 25 (7 u - 1)$

Factor the polynomial by factoring out the greatest common factor, $7 u - 1$ .

$f (x) = - 6 x (x^{2} + 25) + (7 u - 1) (u + 25)$

Step 6

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

$f (x) = - 6 x (x^{2} + 25) + (7 x^{2} - 1) (x^{2} + 25)$

Step 7

Factor $x^{2} + 25$ out of $- 6 x (x^{2} + 25) + (7 x^{2} - 1) (x^{2} + 25)$ .

Factor $x^{2} + 25$ out of $- 6 x (x^{2} + 25)$ .

$f (x) = (x^{2} + 25) (- 6 x) + (7 x^{2} - 1) (x^{2} + 25)$

Factor $x^{2} + 25$ out of $(7 x^{2} - 1) (x^{2} + 25)$ .

$f (x) = (x^{2} + 25) (- 6 x) + (x^{2} + 25) (7 x^{2} - 1)$

Factor $x^{2} + 25$ out of $(x^{2} + 25) (- 6 x) + (x^{2} + 25) (7 x^{2} - 1)$ .

$f (x) = (x^{2} + 25) (- 6 x + 7 x^{2} - 1)$

Step 8

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

$f (x) = (x^{2} + 25) (- 6 u + 7 u^{2} - 1)$

Step 9

Factor by grouping.

Reorder terms.

$f (x) = (x^{2} + 25) (7 u^{2} - 6 u - 1)$

For a polynomial of the form $a x^{2} + i x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 7 \cdot - 1 = - 7$ and whose sum is $i = - 6$ .

Factor $- 6$ out of $- 6 u$ .

$f (x) = (x^{2} + 25) (7 u^{2} - 6 u - 1)$

Rewrite $- 6$ as $1$ plus $- 7$

$f (x) = (x^{2} + 25) (7 u^{2} + (1 - 7) u - 1)$

Apply the distributive property.

$f (x) = (x^{2} + 25) (7 u^{2} + 1 u - 7 u - 1)$

Multiply $u$ by $1$ .

$f (x) = (x^{2} + 25) (7 u^{2} + u - 7 u - 1)$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = (x^{2} + 25) ((7 u^{2} + u) - 7 u - 1)$

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

$f (x) = (x^{2} + 25) (u (7 u + 1) - (7 u + 1))$

Factor the polynomial by factoring out the greatest common factor, $7 u + 1$ .

$f (x) = (x^{2} + 25) ((7 u + 1) (u - 1))$

Step 10

Factor.

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

$f (x) = (x^{2} + 25) ((7 x + 1) (x - 1))$

Remove unnecessary parentheses.

$f (x) = (x^{2} + 25) (7 x + 1) (x - 1)$

Write as a Set of Linear Factors f(x)=7x^4-6x^3+174x^2-150x-25

Related Questions

Support LCMGCF.com by disabling your adblocker.