Basic Math Examples

$1 + a^{7}$

Step 1

Reorder terms.

$a^{7} + 1$

Step 2

Factor $a^{7} + 1$ 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.

${(- 1)}^{7} + 1$

Step 2.3.2

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

$- 1 + 1$

Step 2.3.3

Add $- 1$ and $1$ .

$0$

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.

$\frac{a^{7} + 1}{a + 1}$

Step 2.5

Divide $a^{7} + 1$ 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 $0$ .

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

Step 2.5.2

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

$a^{6}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

Step 2.5.3

Multiply the new quotient term by the divisor.

$a^{6}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

Step 2.5.4

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

$a^{6}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

Step 2.5.5

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

$a^{6}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

Step 2.5.6

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

$a^{6}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

Step 2.5.7

Divide the highest order term in the dividend $- a^{6}$ by the highest order term in divisor $a$ .

$a^{6}$

$a^{5}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

Step 2.5.8

Multiply the new quotient term by the divisor.

$a^{6}$

$a^{5}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

Step 2.5.9

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

$a^{6}$

$a^{5}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

Step 2.5.10

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

$a^{6}$

$a^{5}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

Step 2.5.11

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

$a^{6}$

$a^{5}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

Step 2.5.12

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

$a^{6}$

$a^{5}$

$a^{4}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

Step 2.5.13

Multiply the new quotient term by the divisor.

$a^{6}$

$a^{5}$

$a^{4}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

Step 2.5.14

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

$a^{6}$

$a^{5}$

$a^{4}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

Step 2.5.15

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

$a^{6}$

$a^{5}$

$a^{4}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

Step 2.5.16

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

$a^{6}$

$a^{5}$

$a^{4}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

Step 2.5.17

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

Step 2.5.18

Multiply the new quotient term by the divisor.

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

Step 2.5.19

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

Step 2.5.20

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

Step 2.5.21

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

Step 2.5.22

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

Step 2.5.23

Multiply the new quotient term by the divisor.

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

Step 2.5.24

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

Step 2.5.25

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

Step 2.5.26

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

Step 2.5.27

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

Step 2.5.28

Multiply the new quotient term by the divisor.

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

Step 2.5.29

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

Step 2.5.30

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

Step 2.5.31

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

$1$

Step 2.5.32

Divide the highest order term in the dividend $a$ by the highest order term in divisor $a$ .

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

$1$

Step 2.5.33

Multiply the new quotient term by the divisor.

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

$1$

$a$

$1$

Step 2.5.34

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

$1$

$a$

$1$

Step 2.5.35

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

$a^{6}$

$a^{5}$

$a^{4}$

$a^{3}$

$a^{2}$

$a$

$1$

$a$

$1$

$a^{7}$

$0 a^{6}$

$0 a^{5}$

$0 a^{4}$

$0 a^{3}$

$0 a^{2}$

$0 a$

$1$

$a^{7}$

$a^{6}$

$0 a^{5}$

$a^{6}$

$a^{5}$

$0 a^{4}$

$a^{5}$

$a^{4}$

$0 a^{3}$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$0 a$

$a^{2}$

$a$

$1$

$a$

$1$

$0$

Step 2.5.36

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

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

Step 2.6

Write $a^{7} + 1$ as a set of factors.

$(a + 1) (a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1)$