Pre-Algebra Examples

$(b^{3} \cdot (8 b) + 8) \div (b - 2)$

Step 1

Expand $b^{3} \cdot (8 b) + 8$ .

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Step 1.1

Remove parentheses.

$\frac{b^{3} \cdot (8 b) + 8}{b - 2}$

Step 1.2

Reorder $b^{3}$ and $8$ .

$\frac{8 b^{3} b + 8}{b - 2}$

Step 1.3

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

$\frac{8 (b^{3} b) + 8}{b - 2}$

Step 1.4

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

$\frac{8 b^{3 + 1} + 8}{b - 2}$

Step 1.5

Add $3$ and $1$ .

$\frac{8 b^{4} + 8}{b - 2}$

Step 2

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

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

Step 3

Divide the highest order term in the dividend $8 b^{4}$ by the highest order term in divisor $b$ .

$8 b^{3}$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

Step 4

Multiply the new quotient term by the divisor.

$8 b^{3}$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $8 b^{4} - 16 b^{3}$

$8 b^{3}$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

Step 6

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

$8 b^{3}$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

Step 7

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

$8 b^{3}$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

Step 8

Divide the highest order term in the dividend $16 b^{3}$ by the highest order term in divisor $b$ .

$8 b^{3}$

$16 b^{2}$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

Step 9

Multiply the new quotient term by the divisor.

$8 b^{3}$

$16 b^{2}$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $16 b^{3} - 32 b^{2}$

$8 b^{3}$

$16 b^{2}$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

Step 11

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

$8 b^{3}$

$16 b^{2}$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

Step 12

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

$8 b^{3}$

$16 b^{2}$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

$0 b$

Step 13

Divide the highest order term in the dividend $32 b^{2}$ by the highest order term in divisor $b$ .

$8 b^{3}$

$16 b^{2}$

$32 b$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

$0 b$

Step 14

Multiply the new quotient term by the divisor.

$8 b^{3}$

$16 b^{2}$

$32 b$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

$0 b$

$32 b^{2}$

$64 b$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $32 b^{2} - 64 b$

$8 b^{3}$

$16 b^{2}$

$32 b$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

$0 b$

$32 b^{2}$

$64 b$

Step 16

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

$8 b^{3}$

$16 b^{2}$

$32 b$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

$0 b$

$32 b^{2}$

$64 b$

Step 17

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

$8 b^{3}$

$16 b^{2}$

$32 b$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

$0 b$

$32 b^{2}$

$64 b$

$8$

Step 18

Divide the highest order term in the dividend $64 b$ by the highest order term in divisor $b$ .

$8 b^{3}$

$16 b^{2}$

$32 b$

$64$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

$0 b$

$32 b^{2}$

$64 b$

$8$

Step 19

Multiply the new quotient term by the divisor.

$8 b^{3}$

$16 b^{2}$

$32 b$

$64$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

$0 b$

$32 b^{2}$

$64 b$

$8$

$64 b$

$128$

Step 20

The expression needs to be subtracted from the dividend, so change all the signs in $64 b - 128$

$8 b^{3}$

$16 b^{2}$

$32 b$

$64$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

$0 b$

$32 b^{2}$

$64 b$

$8$

$64 b$

$128$

Step 21

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

$8 b^{3}$

$16 b^{2}$

$32 b$

$64$

$b$

$2$

$8 b^{4}$

$0 b^{3}$

$0 b^{2}$

$0 b$

$8$

$8 b^{4}$

$16 b^{3}$

$0 b^{2}$

$16 b^{3}$

$32 b^{2}$

$0 b$

$32 b^{2}$

$64 b$

$8$

$64 b$

$128$

$136$

Step 22

The final answer is the quotient plus the remainder over the divisor.

$8 b^{3} + 16 b^{2} + 32 b + 64 + \frac{136}{b - 2}$