Write as a Set of Linear Factors f(x)=x^4+14x^3+46x^2-42x-147

$f (x) = x^{4} + 14 x^{3} + 46 x^{2} - 42 x - 147$

Step 1

Regroup terms.

$f (x) = 14 x^{3} - 42 x + x^{4} + 46 x^{2} - 147$

Step 2

Factor $14 x$ out of $14 x^{3} - 42 x$ .

Factor $14 x$ out of $14 x^{3}$ .

$f (x) = 14 x (x^{2}) - 42 x + x^{4} + 46 x^{2} - 147$

Factor $14 x$ out of $- 42 x$ .

$f (x) = 14 x (x^{2}) + 14 x (- 3) + x^{4} + 46 x^{2} - 147$

Factor $14 x$ out of $14 x (x^{2}) + 14 x (- 3)$ .

$f (x) = 14 x (x^{2} - 3) + x^{4} + 46 x^{2} - 147$

Step 3

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

$f (x) = 14 x (x^{2} - 3) + {(x^{2})}^{2} + 46 x^{2} - 147$

Step 4

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

$f (x) = 14 x (x^{2} - 3) + u^{2} + 46 u - 147$

Step 5

Factor $u^{2} + 46 u - 147$ 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 $- 147$ and whose sum is $46$ .

$f (x) = - 3, 49$

Write the factored form using these integers.

$f (x) = 14 x (x^{2} - 3) + (u - 3) (u + 49)$

Step 6

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

$f (x) = 14 x (x^{2} - 3) + (x^{2} - 3) (x^{2} + 49)$

Step 7

Factor $x^{2} - 3$ out of $14 x (x^{2} - 3) + (x^{2} - 3) (x^{2} + 49)$ .

Factor $x^{2} - 3$ out of $14 x (x^{2} - 3)$ .

$f (x) = (x^{2} - 3) (14 x) + (x^{2} - 3) (x^{2} + 49)$

Factor $x^{2} - 3$ out of $(x^{2} - 3) (14 x) + (x^{2} - 3) (x^{2} + 49)$ .

$f (x) = (x^{2} - 3) (14 x + x^{2} + 49)$

Step 8

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

$f (x) = (x^{2} - 3) (14 u + u^{2} + 49)$

Step 9

Factor using the perfect square rule.

Rearrange terms.

$f (x) = (x^{2} - 3) (u^{2} + 14 u + 49)$

Rewrite $49$ as $7^{2}$ .

$f (x) = (x^{2} - 3) (u^{2} + 14 u + 7^{2})$

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

$14 u = 2 \cdot u \cdot 7$

Rewrite the polynomial.

$f (x) = (x^{2} - 3) (u^{2} + 2 \cdot u \cdot 7 + 7^{2})$

Factor using the perfect square trinomial rule $a^{2} + 2 a i + i^{2} = {(a + i)}^{2}$ , where $a = u$ and $i = 7$ .

$f (x) = (x^{2} - 3) {(u + 7)}^{2}$

Step 10

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

$f (x) = (x^{2} - 3) {(x + 7)}^{2}$

Write as a Set of Linear Factors f(x)=x^4+14x^3+46x^2-42x-147

Related Questions

Support LCMGCF.com by disabling your adblocker.