Write as a Set of Linear Factors x^4+6x^3+17x^2+54x+72

$x^{4} + 6 x^{3} + 17 x^{2} + 54 x + 72$

Step 1

Regroup terms.

$6 x^{3} + 54 x + x^{4} + 17 x^{2} + 72$

Step 2

Factor $6 x$ out of $6 x^{3} + 54 x$ .

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

$6 x (x^{2}) + 54 x + x^{4} + 17 x^{2} + 72$

Factor $6 x$ out of $54 x$ .

$6 x (x^{2}) + 6 x (9) + x^{4} + 17 x^{2} + 72$

Factor $6 x$ out of $6 x (x^{2}) + 6 x (9)$ .

$6 x (x^{2} + 9) + x^{4} + 17 x^{2} + 72$

Step 3

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

$6 x (x^{2} + 9) + {(x^{2})}^{2} + 17 x^{2} + 72$

Step 4

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

$6 x (x^{2} + 9) + u^{2} + 17 u + 72$

Step 5

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 $17$ .

$8, 9$

Write the factored form using these integers.

$6 x (x^{2} + 9) + (u + 8) (u + 9)$

Step 6

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

$6 x (x^{2} + 9) + (x^{2} + 8) (x^{2} + 9)$

Step 7

Factor $x^{2} + 9$ out of $6 x (x^{2} + 9) + (x^{2} + 8) (x^{2} + 9)$ .

Factor $x^{2} + 9$ out of $6 x (x^{2} + 9)$ .

$(x^{2} + 9) (6 x) + (x^{2} + 8) (x^{2} + 9)$

Factor $x^{2} + 9$ out of $(x^{2} + 8) (x^{2} + 9)$ .

$(x^{2} + 9) (6 x) + (x^{2} + 9) (x^{2} + 8)$

Factor $x^{2} + 9$ out of $(x^{2} + 9) (6 x) + (x^{2} + 9) (x^{2} + 8)$ .

$(x^{2} + 9) (6 x + x^{2} + 8)$

Step 8

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

$(x^{2} + 9) (6 u + u^{2} + 8)$

Step 9

Factor $6 u + u^{2} + 8$ 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 $6$ .

$2, 4$

Write the factored form using these integers.

$(x^{2} + 9) ((u + 2) (u + 4))$

Step 10

Factor.

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

$(x^{2} + 9) ((x + 2) (x + 4))$

Remove unnecessary parentheses.

$(x^{2} + 9) (x + 2) (x + 4)$

Write as a Set of Linear Factors x^4+6x^3+17x^2+54x+72

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