Factor x^4-9x^2+6x^3-54x-7x^2+63

$x^{4} - 9 x^{2} + 6 x^{3} - 54 x - 7 x^{2} + 63$

Step 1

Regroup terms.

$x^{4} + 6 x^{3} - 7 x^{2} - 9 x^{2} - 54 x + 63$

Step 2

Factor $x^{2}$ out of $x^{4} + 6 x^{3} - 7 x^{2}$ .

Factor $x^{2}$ out of $x^{4}$ .

$x^{2} x^{2} + 6 x^{3} - 7 x^{2} - 9 x^{2} - 54 x + 63$

Factor $x^{2}$ out of $6 x^{3}$ .

$x^{2} x^{2} + x^{2} (6 x) - 7 x^{2} - 9 x^{2} - 54 x + 63$

Factor $x^{2}$ out of $- 7 x^{2}$ .

$x^{2} x^{2} + x^{2} (6 x) + x^{2} \cdot - 7 - 9 x^{2} - 54 x + 63$

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

$x^{2} (x^{2} + 6 x) + x^{2} \cdot - 7 - 9 x^{2} - 54 x + 63$

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

$x^{2} (x^{2} + 6 x - 7) - 9 x^{2} - 54 x + 63$

Step 3

Factor.

Factor $x^{2} + 6 x - 7$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 7$ and whose sum is $6$ .

$- 1, 7$

Write the factored form using these integers.

$x^{2} ((x - 1) (x + 7)) - 9 x^{2} - 54 x + 63$

Remove unnecessary parentheses.

$x^{2} (x - 1) (x + 7) - 9 x^{2} - 54 x + 63$

Step 4

Factor $9$ out of $- 9 x^{2} - 54 x + 63$ .

Factor $9$ out of $- 9 x^{2}$ .

$x^{2} (x - 1) (x + 7) + 9 (- x^{2}) - 54 x + 63$

Factor $9$ out of $- 54 x$ .

$x^{2} (x - 1) (x + 7) + 9 (- x^{2}) + 9 (- 6 x) + 63$

Factor $9$ out of $63$ .

$x^{2} (x - 1) (x + 7) + 9 (- x^{2}) + 9 (- 6 x) + 9 (7)$

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

$x^{2} (x - 1) (x + 7) + 9 (- x^{2} - 6 x) + 9 (7)$

Factor $9$ out of $9 (- x^{2} - 6 x) + 9 (7)$ .

$x^{2} (x - 1) (x + 7) + 9 (- x^{2} - 6 x + 7)$

Step 5

Factor.

Factor by grouping.

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

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

$x^{2} (x - 1) (x + 7) + 9 (- x^{2} - 6 (x) + 7)$

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

$x^{2} (x - 1) (x + 7) + 9 (- x^{2} + (1 - 7) x + 7)$

Apply the distributive property.

$x^{2} (x - 1) (x + 7) + 9 (- x^{2} + 1 x - 7 x + 7)$

Multiply $x$ by $1$ .

$x^{2} (x - 1) (x + 7) + 9 (- x^{2} + x - 7 x + 7)$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x^{2} (x - 1) (x + 7) + 9 ((- x^{2} + x) - 7 x + 7)$

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

$x^{2} (x - 1) (x + 7) + 9 (x (- x + 1) + 7 (- x + 1))$

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$x^{2} (x - 1) (x + 7) + 9 ((- x + 1) (x + 7))$

Remove unnecessary parentheses.

$x^{2} (x - 1) (x + 7) + 9 (- x + 1) (x + 7)$

Step 6

Factor $x + 7$ out of $x^{2} (x - 1) (x + 7) + 9 (- x + 1) (x + 7)$ .

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

$(x + 7) (x^{2} (x - 1)) + 9 (- x + 1) (x + 7)$

Factor $x + 7$ out of $9 (- x + 1) (x + 7)$ .

$(x + 7) (x^{2} (x - 1)) + (x + 7) (9 (- x + 1))$

Factor $x + 7$ out of $(x + 7) (x^{2} (x - 1)) + (x + 7) (9 (- x + 1))$ .

$(x + 7) (x^{2} (x - 1) + 9 (- x + 1))$

Step 7

Apply the distributive property.

$(x + 7) (x^{2} x + x^{2} \cdot - 1 + 9 (- x + 1))$

Step 8

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$(x + 7) (x^{2} x^{1} + x^{2} \cdot - 1 + 9 (- x + 1))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 7) (x^{2 + 1} + x^{2} \cdot - 1 + 9 (- x + 1))$

Add $2$ and $1$ .

$(x + 7) (x^{3} + x^{2} \cdot - 1 + 9 (- x + 1))$

Step 9

Move $- 1$ to the left of $x^{2}$ .

$(x + 7) (x^{3} - 1 x^{2} + 9 (- x + 1))$

Step 10

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$(x + 7) (x^{3} - x^{2} + 9 (- x + 1))$

Step 11

Apply the distributive property.

$(x + 7) (x^{3} - x^{2} + 9 (- x) + 9 \cdot 1)$

Step 12

Multiply $- 1$ by $9$ .

$(x + 7) (x^{3} - x^{2} - 9 x + 9 \cdot 1)$

Step 13

Multiply $9$ by $1$ .

$(x + 7) (x^{3} - x^{2} - 9 x + 9)$

Step 14

Factor.

Rewrite $x^{3} - x^{2} - 9 x + 9$ in a factored form.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + 7) ((x^{3} - x^{2}) - 9 x + 9)$

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

$(x + 7) (x^{2} (x - 1) - 9 (x - 1))$

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

$(x + 7) ((x - 1) (x^{2} - 9))$

Rewrite $9$ as $3^{2}$ .

$(x + 7) ((x - 1) (x^{2} - 3^{2}))$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$(x + 7) ((x - 1) ((x + 3) (x - 3)))$

Remove unnecessary parentheses.

$(x + 7) ((x - 1) (x + 3) (x - 3))$

Remove unnecessary parentheses.

$(x + 7) (x - 1) (x + 3) (x - 3)$

Factor x^4-9x^2+6x^3-54x-7x^2+63

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