Factor over the Complex Numbers x^4+2x^3+22x^2+50x-75

$x^{4} + 2 x^{3} + 22 x^{2} + 50 x - 75$

Step 1

Regroup terms.

$2 x^{3} + 50 x + x^{4} + 22 x^{2} - 75$

Step 2

Factor $2 x$ out of $2 x^{3} + 50 x$ .

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

$2 x (x^{2}) + 50 x + x^{4} + 22 x^{2} - 75$

Factor $2 x$ out of $50 x$ .

$2 x (x^{2}) + 2 x (25) + x^{4} + 22 x^{2} - 75$

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

$2 x (x^{2} + 25) + x^{4} + 22 x^{2} - 75$

Step 3

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

$2 x (x^{2} + 25) + {(x^{2})}^{2} + 22 x^{2} - 75$

Step 4

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

$2 x (x^{2} + 25) + u^{2} + 22 u - 75$

Step 5

Factor $u^{2} + 22 u - 75$ 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 $- 75$ and whose sum is $22$ .

$- 3, 25$

Write the factored form using these integers.

$2 x (x^{2} + 25) + (u - 3) (u + 25)$

Step 6

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

$2 x (x^{2} + 25) + (x^{2} - 3) (x^{2} + 25)$

Step 7

Factor $x^{2} + 25$ out of $2 x (x^{2} + 25) + (x^{2} - 3) (x^{2} + 25)$ .

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

$(x^{2} + 25) (2 x) + (x^{2} - 3) (x^{2} + 25)$

Factor $x^{2} + 25$ out of $(x^{2} - 3) (x^{2} + 25)$ .

$(x^{2} + 25) (2 x) + (x^{2} + 25) (x^{2} - 3)$

Factor $x^{2} + 25$ out of $(x^{2} + 25) (2 x) + (x^{2} + 25) (x^{2} - 3)$ .

$(x^{2} + 25) (2 x + x^{2} - 3)$

Step 8

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

$(x^{2} + 25) (2 u + u^{2} - 3)$

Step 9

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

$- 1, 3$

Write the factored form using these integers.

$(x^{2} + 25) ((u - 1) (u + 3))$

Step 10

Factor.

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

$(x^{2} + 25) ((x - 1) (x + 3))$

Remove unnecessary parentheses.

$(x^{2} + 25) (x - 1) (x + 3)$

Step 11

Multiply the constant in the polynomial $x^{2} + 25$ by $- i^{2}$ where $i^{2}$ is equal to $- 1$ .

$(x^{2} - 25 i^{2}) (x - 1) (x + 3)$

Step 12

Rewrite $25 i^{2}$ as ${(5 i)}^{2}$ .

$(x^{2} - {(5 i)}^{2}) (x - 1) (x + 3)$

Step 13

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 $i = 5 i$ .

$(x + 5 i) (x - (5 i)) (x - 1) (x + 3)$

Step 14

Multiply $5$ by $- 1$ .

$(x + 5 i) (x - 5 i) (x - 1) (x + 3)$

Factor over the Complex Numbers x^4+2x^3+22x^2+50x-75

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