Algebra Examples

$x^{4} - 10 x^{3} + 9 x^{2} - x^{2} + 10 x - 9$

Step 1

Regroup terms.

$x^{4} - 10 x^{3} + 9 x^{2} - x^{2} + 10 x - 9$

Step 2

Factor $x^{2}$ out of $x^{4} - 10 x^{3} + 9 x^{2}$ .

Tap for more steps...

Step 2.1

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

$x^{2} x^{2} - 10 x^{3} + 9 x^{2} - x^{2} + 10 x - 9$

Step 2.2

Factor $x^{2}$ out of $- 10 x^{3}$ .

$x^{2} x^{2} + x^{2} (- 10 x) + 9 x^{2} - x^{2} + 10 x - 9$

Step 2.3

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

$x^{2} x^{2} + x^{2} (- 10 x) + x^{2} \cdot 9 - x^{2} + 10 x - 9$

Step 2.4

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (- 10 x)$ .

$x^{2} (x^{2} - 10 x) + x^{2} \cdot 9 - x^{2} + 10 x - 9$

Step 2.5

Factor $x^{2}$ out of $x^{2} (x^{2} - 10 x) + x^{2} \cdot 9$ .

$x^{2} (x^{2} - 10 x + 9) - x^{2} + 10 x - 9$

Step 3

Factor.

Tap for more steps...

Step 3.1

Factor $x^{2} - 10 x + 9$ using the AC method.

Tap for more steps...

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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Step 3.1.2

Write the factored form using these integers.

$x^{2} ((x - 9) (x - 1)) - x^{2} + 10 x - 9$

Step 3.2

Remove unnecessary parentheses.

$x^{2} (x - 9) (x - 1) - x^{2} + 10 x - 9$

Step 4

Factor by grouping.

Tap for more steps...

Step 4.1

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 $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = 10$ .

Tap for more steps...

Step 4.1.1

Factor $10$ out of $10 x$ .

$x^{2} (x - 9) (x - 1) - x^{2} + 10 (x) - 9$

Step 4.1.2

Rewrite $10$ as $1$ plus $9$

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

Multiply $x$ by $1$ .

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

Step 4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.2.1

Group the first two terms and the last two terms.

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

Step 4.2.2

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

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

Step 4.3

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

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

Step 5

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

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Apply the distributive property.

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

Step 7

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

Tap for more steps...

Step 7.1

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

Tap for more steps...

Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.2

Add $2$ and $1$ .

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

Step 8

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

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

Step 9

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

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

Step 10

Factor.

Tap for more steps...

Step 10.1

Rewrite $x^{3} - x^{2} - x + 1$ in a factored form.

Tap for more steps...

Step 10.1.1

Factor out the greatest common factor from each group.

Tap for more steps...

Step 10.1.1.1

Group the first two terms and the last two terms.

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

Step 10.1.1.2

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

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

Step 10.1.2

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

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

Step 10.1.3

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

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

Step 10.1.4

Factor.

Tap for more steps...

Step 10.1.4.1

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 $b = 1$ .

$(x - 9) ((x - 1) ((x + 1) (x - 1)))$

Step 10.1.4.2

Remove unnecessary parentheses.

$(x - 9) ((x - 1) (x + 1) (x - 1))$

Step 10.1.5

Combine exponents.

Tap for more steps...

Step 10.1.5.1

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

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

Step 10.1.5.2

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

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

Step 10.1.5.3

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

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

Step 10.1.5.4

Add $1$ and $1$ .

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

Step 10.2

Remove unnecessary parentheses.

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