Basic Math Examples

(a^{5} - 3 a^{4} + a^{3} + 2 a - 1) \div (a + 3)

$(a^{5} - 3 a^{4} + a^{3} + 2 a - 1) \div (a + 3)$

Step 1

Regroup terms.

$(a^{5} - 1 - 3 a^{4} + a^{3} + 2 a) \div (a + 3)$

Step 2

Factor

$a^{5} - 1$ using the rational roots test.

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Step 2.1

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 2.2

Find every combination of

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

$\pm 1$

Step 2.3

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 2.3.1

Substitute

$1$ into the polynomial.

$1^{5} - 1$

Step 2.3.2

Raise

$1$ to the power of

$5$ .

$1 - 1$

Step 2.3.3

Subtract

$1$ from

$1$ .

$0$

Step 2.4

Since

$1$ is a known root, divide the polynomial by

$a - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{a^{5} - 1}{a - 1}$

Step 2.5

Divide

$a^{5} - 1$ by

$a - 1$ .

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Step 2.5.1

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

$0$ .

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

Step 2.5.2

Divide the highest order term in the dividend

$a^{5}$ by the highest order term in divisor

$a$ .

$a^{4}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

Step 2.5.3

Multiply the new quotient term by the divisor.

$a^{4}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

Step 2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in

$a^{5} - a^{4}$

$a^{4}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

Step 2.5.5

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

$a^{4}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

Step 2.5.6

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

$a^{4}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

Step 2.5.7

Divide the highest order term in the dividend

$a^{4}$ by the highest order term in divisor

$a$ .

$a^{4}$

$a^{3}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

Step 2.5.8

Multiply the new quotient term by the divisor.

$a^{4}$

$a^{3}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

Step 2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in

$a^{4} - a^{3}$

$a^{4}$

$a^{3}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

Step 2.5.10

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

$a^{4}$

$a^{3}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

Step 2.5.11

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

$a^{4}$

$a^{3}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

Step 2.5.12

Divide the highest order term in the dividend

$a^{3}$ by the highest order term in divisor

$a$ .

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

Step 2.5.13

Multiply the new quotient term by the divisor.

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

Step 2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in

$a^{3} - a^{2}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

Step 2.5.15

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

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

Step 2.5.16

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

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

Step 2.5.17

Divide the highest order term in the dividend

$a^{2}$ by the highest order term in divisor

$a$ .

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

Step 2.5.18

Multiply the new quotient term by the divisor.

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

Step 2.5.19

The expression needs to be subtracted from the dividend, so change all the signs in

$a^{2} - a$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

Step 2.5.20

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

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

Step 2.5.21

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

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

$1$

Step 2.5.22

Divide the highest order term in the dividend

$a$ by the highest order term in divisor

$a$ .

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

$1$

Step 2.5.23

Multiply the new quotient term by the divisor.

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

$1$

$a$

$1$

Step 2.5.24

The expression needs to be subtracted from the dividend, so change all the signs in

$a - 1$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

$1$

$a$

$1$

Step 2.5.25

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

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a$

$1$

$a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

$1$

$a$

$1$

$0$

Step 2.5.26

Since the remander is

$0$ , the final answer is the quotient.

$a^{4} + a^{3} + a^{2} + a + 1$

Step 2.6

Write

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

$((a - 1) (a^{4} + a^{3} + a^{2} + a + 1) - 3 a^{4} + a^{3} + 2 a) \div (a + 3)$

Step 3

Factor

$a$ out of

$- 3 a^{4} + a^{3} + 2 a$ .

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Step 3.1

Factor

$a$ out of

$- 3 a^{4}$ .

$((a - 1) (a^{4} + a^{3} + a^{2} + a + 1) + a (- 3 a^{3}) + a^{3} + 2 a) \div (a + 3)$

Step 3.2

Factor

$a$ out of

$a^{3}$ .

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

Step 3.3

Factor

$a$ out of

$2 a$ .

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

Step 3.4

Factor

$a$ out of

$a (- 3 a^{3}) + a \cdot a^{2}$ .

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

Step 3.5

Factor

$a$ out of

$a (- 3 a^{3} + a^{2}) + a \cdot 2$ .

$((a - 1) (a^{4} + a^{3} + a^{2} + a + 1) + a (- 3 a^{3} + a^{2} + 2)) \div (a + 3)$

Step 4

Factor.

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Step 4.1

Factor

$- 3 a^{3} + a^{2} + 2$ using the rational roots test.

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Step 4.1.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 3$

Step 4.1.2

Find every combination of

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

$\pm 1, \pm 0.\overline{3}, \pm 2, \pm 0.\overline{6}$

Step 4.1.3

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 4.1.3.1

Substitute

$1$ into the polynomial.

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

Step 4.1.3.2

Raise

$1$ to the power of

$3$ .

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

Step 4.1.3.3

Multiply

$- 3$ by

$1$ .

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

Step 4.1.3.4

Raise

$1$ to the power of

$2$ .

$- 3 + 1 + 2$

Step 4.1.3.5

Add

$- 3$ and

$1$ .

$- 2 + 2$

Step 4.1.3.6

Add

$- 2$ and

$2$ .

$0$

Step 4.1.4

Since

$1$ is a known root, divide the polynomial by

$a - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- 3 a^{3} + a^{2} + 2}{a - 1}$

Step 4.1.5

Divide

$- 3 a^{3} + a^{2} + 2$ by

$a - 1$ .

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Step 4.1.5.1

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

$0$ .

$a$

$1$

$3 a^{3}$

$a^{2}$

$0 a$

$2$

Step 4.1.5.2

Divide the highest order term in the dividend

$- 3 a^{3}$ by the highest order term in divisor

$a$ .

				-	$3 a^{2}$
	$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$

Step 4.1.5.3

Multiply the new quotient term by the divisor.

			-	$3 a^{2}$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			-	$3 a^{3}$	+	$3 a^{2}$

Step 4.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in

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

			-	$3 a^{2}$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			+	$3 a^{3}$	-	$3 a^{2}$

Step 4.1.5.5

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

			-	$3 a^{2}$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			+	$3 a^{3}$	-	$3 a^{2}$
					-	$2 a^{2}$

Step 4.1.5.6

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

			-	$3 a^{2}$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			+	$3 a^{3}$	-	$3 a^{2}$
					-	$2 a^{2}$	+	$0 a$

Step 4.1.5.7

Divide the highest order term in the dividend

$- 2 a^{2}$ by the highest order term in divisor

$a$ .

			-	$3 a^{2}$	-	$2 a$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			+	$3 a^{3}$	-	$3 a^{2}$
					-	$2 a^{2}$	+	$0 a$

Step 4.1.5.8

Multiply the new quotient term by the divisor.

			-	$3 a^{2}$	-	$2 a$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			+	$3 a^{3}$	-	$3 a^{2}$
					-	$2 a^{2}$	+	$0 a$
					-	$2 a^{2}$	+	$2 a$

Step 4.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in

$- 2 a^{2} + 2 a$

			-	$3 a^{2}$	-	$2 a$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			+	$3 a^{3}$	-	$3 a^{2}$
					-	$2 a^{2}$	+	$0 a$
					+	$2 a^{2}$	-	$2 a$

Step 4.1.5.10

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

			-	$3 a^{2}$	-	$2 a$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			+	$3 a^{3}$	-	$3 a^{2}$
					-	$2 a^{2}$	+	$0 a$
					+	$2 a^{2}$	-	$2 a$
							-	$2 a$

Step 4.1.5.11

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

			-	$3 a^{2}$	-	$2 a$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			+	$3 a^{3}$	-	$3 a^{2}$
					-	$2 a^{2}$	+	$0 a$
					+	$2 a^{2}$	-	$2 a$
							-	$2 a$	+	$2$

Step 4.1.5.12

Divide the highest order term in the dividend

$- 2 a$ by the highest order term in divisor

$a$ .

			-	$3 a^{2}$	-	$2 a$	-	$2$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			+	$3 a^{3}$	-	$3 a^{2}$
					-	$2 a^{2}$	+	$0 a$
					+	$2 a^{2}$	-	$2 a$
							-	$2 a$	+	$2$

Step 4.1.5.13

Multiply the new quotient term by the divisor.

			-	$3 a^{2}$	-	$2 a$	-	$2$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			+	$3 a^{3}$	-	$3 a^{2}$
					-	$2 a^{2}$	+	$0 a$
					+	$2 a^{2}$	-	$2 a$
							-	$2 a$	+	$2$
							-	$2 a$	+	$2$

Step 4.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in

$- 2 a + 2$

			-	$3 a^{2}$	-	$2 a$	-	$2$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			+	$3 a^{3}$	-	$3 a^{2}$
					-	$2 a^{2}$	+	$0 a$
					+	$2 a^{2}$	-	$2 a$
							-	$2 a$	+	$2$
							+	$2 a$	-	$2$

Step 4.1.5.15

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

			-	$3 a^{2}$	-	$2 a$	-	$2$
$a$	-	$1$	-	$3 a^{3}$	+	$a^{2}$	+	$0 a$	+	$2$
			+	$3 a^{3}$	-	$3 a^{2}$
					-	$2 a^{2}$	+	$0 a$
					+	$2 a^{2}$	-	$2 a$
							-	$2 a$	+	$2$
							+	$2 a$	-	$2$
										$0$

Step 4.1.5.16

Since the remander is

$0$ , the final answer is the quotient.

$- 3 a^{2} - 2 a - 2$

Step 4.1.6

Write

$- 3 a^{3} + a^{2} + 2$ as a set of factors.

$((a - 1) (a^{4} + a^{3} + a^{2} + a + 1) + a ((a - 1) (- 3 a^{2} - 2 a - 2))) \div (a + 3)$

Step 4.2

Remove unnecessary parentheses.

$((a - 1) (a^{4} + a^{3} + a^{2} + a + 1) + a (a - 1) (- 3 a^{2} - 2 a - 2)) \div (a + 3)$

Step 5

Factor

$a - 1$ out of

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

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Step 5.1

Factor

$a - 1$ out of

$a (a - 1) (- 3 a^{2} - 2 a - 2)$ .

$((a - 1) (a^{4} + a^{3} + a^{2} + a + 1) + (a - 1) (a (- 3 a^{2} - 2 a - 2))) \div (a + 3)$

Step 5.2

Factor

$a - 1$ out of

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

$(a - 1) (a^{4} + a^{3} + a^{2} + a + 1 + a (- 3 a^{2} - 2 a - 2)) \div (a + 3)$

Step 6

Apply the distributive property.

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

Step 7

Simplify.

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Step 7.1

Rewrite using the commutative property of multiplication.

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

Step 7.2

Rewrite using the commutative property of multiplication.

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

Step 7.3

Move

$- 2$ to the left of

$a$ .

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

Step 8

Simplify each term.

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Step 8.1

Multiply

$a$ by

$a^{2}$ by adding the exponents.

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Step 8.1.1

Move

$a^{2}$ .

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

Step 8.1.2

Multiply

$a^{2}$ by

$a$ .

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Step 8.1.2.1

Raise

$a$ to the power of

$1$ .

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

Step 8.1.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8.1.3

Add

$2$ and

$1$ .

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

Step 8.2

Multiply

$a$ by

$a$ by adding the exponents.

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Step 8.2.1

Move

$a$ .

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

Step 8.2.2

Multiply

$a$ by

$a$ .

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

$(a - 1) (a^{4} + a^{3} + a^{2} + a + 1 - 3 a^{3} - 2 a^{2} - 2 a) \div (a + 3)$

Step 9

Subtract

$3 a^{3}$ from

$a^{3}$ .

$(a - 1) (a^{4} - 2 a^{3} + a^{2} + a + 1 - 2 a^{2} - 2 a) \div (a + 3)$

Step 10

Subtract

$2 a^{2}$ from

$a^{2}$ .

$(a - 1) (a^{4} - 2 a^{3} - a^{2} + a + 1 - 2 a) \div (a + 3)$

Step 11

Subtract

$2 a$ from

$a$ .

$(a - 1) (a^{4} - 2 a^{3} - a^{2} - a + 1) \div (a + 3)$