Pre-Algebra Examples

$\frac{1}{4} x^{4} + \frac{3}{4} x^{8} - \frac{5}{4} x^{6} + \frac{1}{4} x$

Step 1

Factor $x$ out of $\frac{1}{4} x^{4} + \frac{3}{4} x^{8} - \frac{5}{4} x^{6} + \frac{1}{4} x$ .

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Step 1.1

Factor $x$ out of $\frac{1}{4} x^{4}$ .

$x (\frac{1}{4} x^{3}) + \frac{3}{4} x^{8} - \frac{5}{4} x^{6} + \frac{1}{4} x$

Step 1.2

Factor $x$ out of $\frac{3}{4} x^{8}$ .

$x (\frac{1}{4} x^{3}) + x (\frac{3}{4} x^{7}) - \frac{5}{4} x^{6} + \frac{1}{4} x$

Step 1.3

Factor $x$ out of $- \frac{5}{4} x^{6}$ .

$x (\frac{1}{4} x^{3}) + x (\frac{3}{4} x^{7}) + x (- \frac{5}{4} x^{5}) + \frac{1}{4} x$

Step 1.4

Factor $x$ out of $\frac{1}{4} x$ .

$x (\frac{1}{4} x^{3}) + x (\frac{3}{4} x^{7}) + x (- \frac{5}{4} x^{5}) + x \frac{1}{4}$

Step 1.5

Factor $x$ out of $x (\frac{1}{4} x^{3}) + x (\frac{3}{4} x^{7})$ .

$x (\frac{1}{4} x^{3} + \frac{3}{4} x^{7}) + x (- \frac{5}{4} x^{5}) + x \frac{1}{4}$

Step 1.6

Factor $x$ out of $x (\frac{1}{4} x^{3} + \frac{3}{4} x^{7}) + x (- \frac{5}{4} x^{5})$ .

$x (\frac{1}{4} x^{3} + \frac{3}{4} x^{7} - \frac{5}{4} x^{5}) + x \frac{1}{4}$

Step 1.7

Factor $x$ out of $x (\frac{1}{4} x^{3} + \frac{3}{4} x^{7} - \frac{5}{4} x^{5}) + x \frac{1}{4}$ .

$x (\frac{1}{4} x^{3} + \frac{3}{4} x^{7} - \frac{5}{4} x^{5} + \frac{1}{4})$

Step 2

Combine $\frac{1}{4}$ and $x^{3}$ .

$x (\frac{x^{3}}{4} + \frac{3}{4} x^{7} - \frac{5}{4} x^{5} + \frac{1}{4})$

Step 3

Combine $\frac{3}{4}$ and $x^{7}$ .

$x (\frac{x^{3}}{4} + \frac{3 x^{7}}{4} - \frac{5}{4} x^{5} + \frac{1}{4})$

Step 4

Combine $x^{5}$ and $\frac{5}{4}$ .

$x (\frac{x^{3}}{4} + \frac{3 x^{7}}{4} - \frac{x^{5} \cdot 5}{4} + \frac{1}{4})$

Step 5

Move $5$ to the left of $x^{5}$ .

$x (\frac{x^{3}}{4} + \frac{3 x^{7}}{4} - \frac{5 \cdot x^{5}}{4} + \frac{1}{4})$

Step 6

Multiply $5$ by $x^{5}$ .

$x (\frac{x^{3}}{4} + \frac{3 x^{7}}{4} - \frac{5 x^{5}}{4} + \frac{1}{4})$

Step 7

Reorder terms.

$x (\frac{3 x^{7}}{4} - \frac{5 x^{5}}{4} + \frac{x^{3}}{4} + \frac{1}{4})$

Step 8

Factor.

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Step 8.1

Rewrite $\frac{3 x^{7}}{4} - \frac{5 x^{5}}{4} + \frac{x^{3}}{4} + \frac{1}{4}$ in a factored form.

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Step 8.1.1

Regroup terms.

$x (\frac{3 x^{7}}{4} - \frac{5 x^{5}}{4} + \frac{x^{3}}{4} + \frac{1}{4})$

Step 8.1.2

Factor $\frac{1}{4}$ out of each term.

$x (\frac{1}{4} (3 x^{7} - (5 x^{5})) + \frac{x^{3}}{4} + \frac{1}{4})$

Step 8.1.3

Factor $\frac{1}{4}$ out of each term.

$x (\frac{1}{4} (3 x^{7} - (5 x^{5})) + \frac{1}{4} (x^{3} + 1))$

Step 8.1.4

Remove parentheses.

$x (\frac{1}{4} (3 x^{7} - 1 \cdot 5 x^{5}) + \frac{1}{4} (x^{3} + 1))$

Step 8.1.5

Factor $\frac{1}{4}$ out of $\frac{1}{4} (3 x^{7} - 1 \cdot 5 x^{5}) + \frac{1}{4} (x^{3} + 1)$ .

$x (\frac{1}{4} (3 x^{7} - 1 \cdot 5 x^{5} + x^{3} + 1))$

Step 8.1.6

Multiply $- 1$ by $5$ .

$x (\frac{1}{4} (3 x^{7} - 5 x^{5} + x^{3} + 1))$

Step 8.1.7

Factor.

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Step 8.1.7.1

Factor $3 x^{7} - 5 x^{5} + x^{3} + 1$ using the rational roots test.

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Step 8.1.7.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1$

$q = \pm 1, \pm 3$

Step 8.1.7.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.\overline{3}$

Step 8.1.7.1.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 8.1.7.1.3.1

Substitute $1$ into the polynomial.

$3 \cdot 1^{7} - 5 \cdot 1^{5} + 1^{3} + 1$

Step 8.1.7.1.3.2

Raise $1$ to the power of $7$ .

$3 \cdot 1 - 5 \cdot 1^{5} + 1^{3} + 1$

Step 8.1.7.1.3.3

Multiply $3$ by $1$ .

$3 - 5 \cdot 1^{5} + 1^{3} + 1$

Step 8.1.7.1.3.4

Raise $1$ to the power of $5$ .

$3 - 5 \cdot 1 + 1^{3} + 1$

Step 8.1.7.1.3.5

Multiply $- 5$ by $1$ .

$3 - 5 + 1^{3} + 1$

Step 8.1.7.1.3.6

Subtract $5$ from $3$ .

$- 2 + 1^{3} + 1$

Step 8.1.7.1.3.7

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

$- 2 + 1 + 1$

Step 8.1.7.1.3.8

Add $- 2$ and $1$ .

$- 1 + 1$

Step 8.1.7.1.3.9

Add $- 1$ and $1$ .

$0$

Step 8.1.7.1.4

Since $1$ is a known root, divide the polynomial by $x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{7} - 5 x^{5} + x^{3} + 1}{x - 1}$

Step 8.1.7.1.5

Divide $3 x^{7} - 5 x^{5} + x^{3} + 1$ by $x - 1$ .

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Step 8.1.7.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 8.1.7.1.5.2

Divide the highest order term in the dividend $3 x^{7}$ by the highest order term in divisor $x$ .

$3 x^{6}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 8.1.7.1.5.3

Multiply the new quotient term by the divisor.

$3 x^{6}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

Step 8.1.7.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{7} - 3 x^{6}$

$3 x^{6}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

Step 8.1.7.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x^{6}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

Step 8.1.7.1.5.6

Pull the next terms from the original dividend down into the current dividend.

$3 x^{6}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

Step 8.1.7.1.5.7

Divide the highest order term in the dividend $3 x^{6}$ by the highest order term in divisor $x$ .

$3 x^{6}$

$3 x^{5}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

Step 8.1.7.1.5.8

Multiply the new quotient term by the divisor.

$3 x^{6}$

$3 x^{5}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

Step 8.1.7.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{6} - 3 x^{5}$

$3 x^{6}$

$3 x^{5}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

Step 8.1.7.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x^{6}$

$3 x^{5}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

Step 8.1.7.1.5.11

Pull the next terms from the original dividend down into the current dividend.

$3 x^{6}$

$3 x^{5}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

Step 8.1.7.1.5.12

Divide the highest order term in the dividend $- 2 x^{5}$ by the highest order term in divisor $x$ .

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

Step 8.1.7.1.5.13

Multiply the new quotient term by the divisor.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

Step 8.1.7.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{5} + 2 x^{4}$

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

Step 8.1.7.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

Step 8.1.7.1.5.16

Pull the next terms from the original dividend down into the current dividend.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

Step 8.1.7.1.5.17

Divide the highest order term in the dividend $- 2 x^{4}$ by the highest order term in divisor $x$ .

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

Step 8.1.7.1.5.18

Multiply the new quotient term by the divisor.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

Step 8.1.7.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{4} + 2 x^{3}$

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

Step 8.1.7.1.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

Step 8.1.7.1.5.21

Pull the next terms from the original dividend down into the current dividend.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

Step 8.1.7.1.5.22

Divide the highest order term in the dividend $- x^{3}$ by the highest order term in divisor $x$ .

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

Step 8.1.7.1.5.23

Multiply the new quotient term by the divisor.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 8.1.7.1.5.24

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{3} + x^{2}$

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 8.1.7.1.5.25

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 8.1.7.1.5.26

Pull the next terms from the original dividend down into the current dividend.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

Step 8.1.7.1.5.27

Divide the highest order term in the dividend $- x^{2}$ by the highest order term in divisor $x$ .

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

Step 8.1.7.1.5.28

Multiply the new quotient term by the divisor.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 8.1.7.1.5.29

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{2} + x$

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 8.1.7.1.5.30

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 8.1.7.1.5.31

Pull the next terms from the original dividend down into the current dividend.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

Step 8.1.7.1.5.32

Divide the highest order term in the dividend $- x$ by the highest order term in divisor $x$ .

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

Step 8.1.7.1.5.33

Multiply the new quotient term by the divisor.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

Step 8.1.7.1.5.34

The expression needs to be subtracted from the dividend, so change all the signs in $- x + 1$

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

Step 8.1.7.1.5.35

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x^{6}$

$3 x^{5}$

$2 x^{4}$

$2 x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$3 x^{7}$

$0 x^{6}$

$5 x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$3 x^{7}$

$3 x^{6}$

$5 x^{5}$

$3 x^{6}$

$3 x^{5}$

$2 x^{5}$

$0 x^{4}$

$2 x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

$0$

Step 8.1.7.1.5.36

Since the remander is $0$ , the final answer is the quotient.

$3 x^{6} + 3 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - x - 1$

Step 8.1.7.1.6

Write $3 x^{7} - 5 x^{5} + x^{3} + 1$ as a set of factors.

$x (\frac{1}{4} ((x - 1) (3 x^{6} + 3 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - x - 1)))$

Step 8.1.7.2

Remove unnecessary parentheses.

$x (\frac{1}{4} (x - 1) (3 x^{6} + 3 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - x - 1))$

Step 8.2

Remove unnecessary parentheses.

$x \frac{1}{4} (x - 1) (3 x^{6} + 3 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - x - 1)$

Step 9

Combine $x$ and $\frac{1}{4}$ .

$\frac{x}{4} (x - 1) (3 x^{6} + 3 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - x - 1)$