Algebra Examples

$(2 x^{4} + 3 x^{3} + 7 x^{2} + 7 x + 6)$ by $(2 x + 1)$

Step 1

Write the problem as a mathematical expression.

$\frac{(2 x^{4} + 3 x^{3} + 7 x^{2} + 7 x + 6)}{(2 x + 1)}$

Step 2

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

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

Step 3

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

Step 4

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

Step 5

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

Step 6

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

Step 7

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

Step 8

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

Step 9

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

Step 10

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

$x^{3}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

Step 11

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

$x^{3}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

Step 12

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

$x^{3}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

Step 13

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$x^{2}$

$3 x$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

Step 14

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$3 x$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} + 3 x$

$x^{3}$

$x^{2}$

$3 x$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

Step 16

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

$x^{3}$

$x^{2}$

$3 x$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

$4 x$

Step 17

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

$x^{3}$

$x^{2}$

$3 x$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

$4 x$

$6$

Step 18

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $2 x$ .

$x^{3}$

$x^{2}$

$3 x$

$2$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

$4 x$

$6$

Step 19

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$3 x$

$2$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

$4 x$

$6$

$4 x$

$2$

Step 20

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

$x^{3}$

$x^{2}$

$3 x$

$2$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

$4 x$

$6$

$4 x$

$2$

Step 21

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

$x^{3}$

$x^{2}$

$3 x$

$2$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

$4 x$

$6$

$4 x$

$2$

$4$

Step 22

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

$x^{3} + x^{2} + 3 x + 2 + \frac{4}{2 x + 1}$