Algebra Examples

$- 3 + 3 x^{5}$

Step 1

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

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Step 1.1

Reorder $- 3$ and $3 x^{5}$ .

$3 x^{5} - 3$

Step 1.2

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

$- 3 (- x^{5}) - 3$

Step 1.3

Factor $- 3$ out of $- 3$ .

$- 3 (- x^{5}) - 3 (1)$

Step 1.4

Factor $- 3$ out of $- 3 (- x^{5}) - 3 (1)$ .

$- 3 (- x^{5} + 1)$

Step 2

Factor.

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Step 2.1

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

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Step 2.1.1

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.

$p = \pm 1$

$q = \pm 1$

Step 2.1.2

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

$\pm 1$

Step 2.1.3

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 2.1.3.1

Substitute $1$ into the polynomial.

$- 1^{5} + 1$

Step 2.1.3.2

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

$- 1 \cdot 1 + 1$

Step 2.1.3.3

Multiply $- 1$ by $1$ .

$- 1 + 1$

Step 2.1.3.4

Add $- 1$ and $1$ .

$0$

Step 2.1.4

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.

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

Step 2.1.5

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

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Step 2.1.5.1

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

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 2.1.5.2

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

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

Step 2.1.5.4

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

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

Step 2.1.5.5

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

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

Step 2.1.5.6

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

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

Step 2.1.5.7

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

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

Step 2.1.5.9

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

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

Step 2.1.5.10

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

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

Step 2.1.5.11

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

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 2.1.5.12

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 2.1.5.14

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 2.1.5.15

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 2.1.5.16

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

Step 2.1.5.17

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

Step 2.1.5.18

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 2.1.5.19

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 2.1.5.20

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 2.1.5.21

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

Step 2.1.5.22

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

Step 2.1.5.23

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

Step 2.1.5.24

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

Step 2.1.5.25

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

$0$

Step 2.1.5.26

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

$- x^{4} - x^{3} - x^{2} - x - 1$

Step 2.1.6

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

$- 3 ((x - 1) (- x^{4} - x^{3} - x^{2} - x - 1))$

Step 2.2

Remove unnecessary parentheses.

$- 3 (x - 1) (- x^{4} - x^{3} - x^{2} - x - 1)$

Step 3

Factor.

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Step 3.1

Factor $- 1$ out of $- x^{4} - x^{3} - x^{2} - x - 1$ .

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Step 3.1.1

Factor $- 1$ out of $- x^{4}$ .

$- 3 (x - 1) (- (x^{4}) - x^{3} - x^{2} - x - 1)$

Step 3.1.2

Factor $- 1$ out of $- x^{3}$ .

$- 3 (x - 1) (- (x^{4}) - (x^{3}) - x^{2} - x - 1)$

Step 3.1.3

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

$- 3 (x - 1) (- (x^{4}) - (x^{3}) - (x^{2}) - x - 1)$

Step 3.1.4

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

$- 3 (x - 1) (- (x^{4}) - (x^{3}) - (x^{2}) - (x) - 1)$

Step 3.1.5

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

$- 3 (x - 1) (- (x^{4}) - (x^{3}) - (x^{2}) - (x) - 1 (1))$

Step 3.1.6

Factor $- 1$ out of $- (x^{4}) - (x^{3})$ .

$- 3 (x - 1) (- (x^{4} + x^{3}) - (x^{2}) - (x) - 1 (1))$

Step 3.1.7

Factor $- 1$ out of $- (x^{4} + x^{3}) - (x^{2})$ .

$- 3 (x - 1) (- (x^{4} + x^{3} + x^{2}) - (x) - 1 (1))$

Step 3.1.8

Factor $- 1$ out of $- (x^{4} + x^{3} + x^{2}) - (x)$ .

$- 3 (x - 1) (- (x^{4} + x^{3} + x^{2} + x) - 1 (1))$

Step 3.1.9

Factor $- 1$ out of $- (x^{4} + x^{3} + x^{2} + x) - 1 (1)$ .

$- 3 (x - 1) (- (x^{4} + x^{3} + x^{2} + x + 1))$

Step 3.2

Remove unnecessary parentheses.

$- 3 (x - 1) \cdot - 1 (x^{4} + x^{3} + x^{2} + x + 1)$

Step 4

Combine exponents.

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Step 4.1

Factor out negative.

$- (- 3 (x - 1) (x^{4} + x^{3} + x^{2} + x + 1))$

Step 4.2

Multiply $- 3$ by $- 1$ .

$3 ((x - 1) (x^{4} + x^{3} + x^{2} + x + 1))$

Step 5

Remove unnecessary parentheses.

$3 (x - 1) (x^{4} + x^{3} + x^{2} + x + 1)$