Algebra Examples

$x^{4} - 9 x^{2} - 5 x^{3} + 45 x + 6 x^{2} - 54$

Step 1

Regroup terms.

$x^{4} - 5 x^{3} + 6 x^{2} - 9 x^{2} + 45 x - 54$

Step 2

Factor $x^{2}$ out of $x^{4} - 5 x^{3} + 6 x^{2}$ .

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Step 2.1

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

$x^{2} x^{2} - 5 x^{3} + 6 x^{2} - 9 x^{2} + 45 x - 54$

Step 2.2

Factor $x^{2}$ out of $- 5 x^{3}$ .

$x^{2} x^{2} + x^{2} (- 5 x) + 6 x^{2} - 9 x^{2} + 45 x - 54$

Step 2.3

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

$x^{2} x^{2} + x^{2} (- 5 x) + x^{2} \cdot 6 - 9 x^{2} + 45 x - 54$

Step 2.4

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (- 5 x)$ .

$x^{2} (x^{2} - 5 x) + x^{2} \cdot 6 - 9 x^{2} + 45 x - 54$

Step 2.5

Factor $x^{2}$ out of $x^{2} (x^{2} - 5 x) + x^{2} \cdot 6$ .

$x^{2} (x^{2} - 5 x + 6) - 9 x^{2} + 45 x - 54$

Step 3

Factor.

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Step 3.1

Factor $x^{2} - 5 x + 6$ using the AC method.

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Step 3.1.1

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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 3.1.2

Write the factored form using these integers.

$x^{2} ((x - 3) (x - 2)) - 9 x^{2} + 45 x - 54$

Step 3.2

Remove unnecessary parentheses.

$x^{2} (x - 3) (x - 2) - 9 x^{2} + 45 x - 54$

Step 4

Factor $9$ out of $- 9 x^{2} + 45 x - 54$ .

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Step 4.1

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

$x^{2} (x - 3) (x - 2) + 9 (- x^{2}) + 45 x - 54$

Step 4.2

Factor $9$ out of $45 x$ .

$x^{2} (x - 3) (x - 2) + 9 (- x^{2}) + 9 (5 x) - 54$

Step 4.3

Factor $9$ out of $- 54$ .

$x^{2} (x - 3) (x - 2) + 9 (- x^{2}) + 9 (5 x) + 9 (- 6)$

Step 4.4

Factor $9$ out of $9 (- x^{2}) + 9 (5 x)$ .

$x^{2} (x - 3) (x - 2) + 9 (- x^{2} + 5 x) + 9 (- 6)$

Step 4.5

Factor $9$ out of $9 (- x^{2} + 5 x) + 9 (- 6)$ .

$x^{2} (x - 3) (x - 2) + 9 (- x^{2} + 5 x - 6)$

Step 5

Factor.

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Step 5.1

Factor by grouping.

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Step 5.1.1

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 $a \cdot c = - 1 \cdot - 6 = 6$ and whose sum is $b = 5$ .

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Step 5.1.1.1

Factor $5$ out of $5 x$ .

$x^{2} (x - 3) (x - 2) + 9 (- x^{2} + 5 (x) - 6)$

Step 5.1.1.2

Rewrite $5$ as $2$ plus $3$

$x^{2} (x - 3) (x - 2) + 9 (- x^{2} + (2 + 3) x - 6)$

Step 5.1.1.3

Apply the distributive property.

$x^{2} (x - 3) (x - 2) + 9 (- x^{2} + 2 x + 3 x - 6)$

Step 5.1.2

Factor out the greatest common factor from each group.

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Step 5.1.2.1

Group the first two terms and the last two terms.

$x^{2} (x - 3) (x - 2) + 9 ((- x^{2} + 2 x) + 3 x - 6)$

Step 5.1.2.2

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

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

Step 5.1.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

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

Step 5.2

Remove unnecessary parentheses.

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

Step 6

Factor $x - 3$ out of $x^{2} (x - 3) (x - 2) + 9 (- x + 2) (x - 3)$ .

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Step 6.1

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

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

Step 6.2

Factor $x - 3$ out of $9 (- x + 2) (x - 3)$ .

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

Step 6.3

Factor $x - 3$ out of $(x - 3) (x^{2} (x - 2)) + (x - 3) (9 (- x + 2))$ .

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

Step 7

Apply the distributive property.

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

Step 8

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

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Step 8.1

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

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Step 8.1.1

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

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

Step 8.1.2

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

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

Step 8.2

Add $2$ and $1$ .

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

Step 9

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

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

Step 10

Apply the distributive property.

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

Step 11

Multiply $- 1$ by $9$ .

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

Step 12

Multiply $9$ by $2$ .

$(x - 3) (x^{3} - 2 x^{2} - 9 x + 18)$

Step 13

Factor.

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Step 13.1

Rewrite $x^{3} - 2 x^{2} - 9 x + 18$ in a factored form.

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Step 13.1.1

Factor out the greatest common factor from each group.

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Step 13.1.1.1

Group the first two terms and the last two terms.

$(x - 3) ((x^{3} - 2 x^{2}) - 9 x + 18)$

Step 13.1.1.2

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

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

Step 13.1.2

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

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

Step 13.1.3

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

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

Step 13.1.4

Factor.

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Step 13.1.4.1

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 $b = 3$ .

$(x - 3) ((x - 2) ((x + 3) (x - 3)))$

Step 13.1.4.2

Remove unnecessary parentheses.

$(x - 3) ((x - 2) (x + 3) (x - 3))$

Step 13.2

Remove unnecessary parentheses.

$(x - 3) (x - 2) (x + 3) (x - 3)$

Step 14

Combine exponents.

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Step 14.1

Raise $x - 3$ to the power of $1$ .

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

Step 14.2

Raise $x - 3$ to the power of $1$ .

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

Step 14.3

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

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

Step 14.4

Add $1$ and $1$ .

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