Algebra Examples

$(6 x^{4} + 15 x^{2} - 8 - 20 x^{3}) \div (- x^{2} + 2 x - 1)$

Step 1

Rewrite the division as a fraction.

$\frac{6 x^{4} + 15 x^{2} - 8 - 20 x^{3}}{- x^{2} + 2 x - 1}$

Step 2

Reorder terms.

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{- x^{2} + 2 x - 1}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = 2$ .

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Step 3.1.1

Factor $2$ out of $2 x$ .

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{- x^{2} + 2 (x) - 1}$

Step 3.1.2

Rewrite $2$ as $1$ plus $1$

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{- x^{2} + (1 + 1) x - 1}$

Step 3.1.3

Apply the distributive property.

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{- x^{2} + 1 x + 1 x - 1}$

Step 3.1.4

Multiply $x$ by $1$ .

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{- x^{2} + x + 1 x - 1}$

Step 3.1.5

Multiply $x$ by $1$ .

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{- x^{2} + x + x - 1}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{(- x^{2} + x) + x - 1}$

Step 3.2.2

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

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{x (- x + 1) - 1 (- x + 1)}$

Step 3.3

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

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{(- x + 1) (x - 1)}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $- 1$ out of $- x$ .

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{(- (x) + 1) (x - 1)}$

Step 4.2

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{(- (x) - 1 (- 1)) (x - 1)}$

Step 4.3

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

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{- (x - 1) (x - 1)}$

Step 4.4

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{- 1 (x - 1) (x - 1)}$

Step 4.5

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

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{- 1 ({(x - 1)}^{1} (x - 1))}$

Step 4.6

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

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{- 1 ({(x - 1)}^{1} {(x - 1)}^{1})}$

Step 4.7

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

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{- 1 {(x - 1)}^{1 + 1}}$

Step 4.8

Add $1$ and $1$ .

$\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{- 1 {(x - 1)}^{2}}$

Step 5

Move the negative in front of the fraction.

$- \frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{{(x - 1)}^{2}}$

Step 6

Split the fraction $\frac{6 x^{4} - 20 x^{3} + 15 x^{2} - 8}{{(x - 1)}^{2}}$ into two fractions.

$- (\frac{6 x^{4} - 20 x^{3} + 15 x^{2}}{{(x - 1)}^{2}} + \frac{- 8}{{(x - 1)}^{2}})$

Step 7

Split the fraction $\frac{6 x^{4} - 20 x^{3} + 15 x^{2}}{{(x - 1)}^{2}}$ into two fractions.

$- (\frac{6 x^{4} - 20 x^{3}}{{(x - 1)}^{2}} + \frac{15 x^{2}}{{(x - 1)}^{2}} + \frac{- 8}{{(x - 1)}^{2}})$

Step 8

Split the fraction $\frac{6 x^{4} - 20 x^{3}}{{(x - 1)}^{2}}$ into two fractions.

$- (\frac{6 x^{4}}{{(x - 1)}^{2}} + \frac{- 20 x^{3}}{{(x - 1)}^{2}} + \frac{15 x^{2}}{{(x - 1)}^{2}} + \frac{- 8}{{(x - 1)}^{2}})$

Step 9

Move the negative in front of the fraction.

$- (\frac{6 x^{4}}{{(x - 1)}^{2}} - \frac{20 x^{3}}{{(x - 1)}^{2}} + \frac{15 x^{2}}{{(x - 1)}^{2}} + \frac{- 8}{{(x - 1)}^{2}})$

Step 10

Move the negative in front of the fraction.

$- (\frac{6 x^{4}}{{(x - 1)}^{2}} - \frac{20 x^{3}}{{(x - 1)}^{2}} + \frac{15 x^{2}}{{(x - 1)}^{2}} - \frac{8}{{(x - 1)}^{2}})$