Divide Using Long Polynomial Division (3x^4-8x^3+9x+5)÷(x-2)

$(3 x^{4} - 8 x^{3} + 9 x + 5) \div (x - 2)$

Step 1

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

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

Step 2

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

Step 3

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} - 6 x^{3}$

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

Step 5

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

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

Step 6

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

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

Step 7

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

$3 x^{3}$

$2 x^{2}$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$3 x^{3}$

$2 x^{2}$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

Step 9

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

$3 x^{3}$

$2 x^{2}$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

Step 10

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

$3 x^{3}$

$2 x^{2}$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

Step 11

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

$3 x^{3}$

$2 x^{2}$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$9 x$

Step 12

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

$3 x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$9 x$

Step 13

Multiply the new quotient term by the divisor.

$3 x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$9 x$

$4 x^{2}$

$8 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 8 x$

$3 x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$9 x$

$4 x^{2}$

$8 x$

Step 15

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

$3 x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$9 x$

$4 x^{2}$

$8 x$

$x$

Step 16

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

$3 x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$9 x$

$4 x^{2}$

$8 x$

$x$

$5$

Step 17

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

$3 x^{3}$

$2 x^{2}$

$4 x$

$1$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$9 x$

$4 x^{2}$

$8 x$

$x$

$5$

Step 18

Multiply the new quotient term by the divisor.

$3 x^{3}$

$2 x^{2}$

$4 x$

$1$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$9 x$

$4 x^{2}$

$8 x$

$x$

$5$

$x$

$2$

Step 19

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

$3 x^{3}$

$2 x^{2}$

$4 x$

$1$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$9 x$

$4 x^{2}$

$8 x$

$x$

$5$

$x$

$2$

Step 20

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

$3 x^{3}$

$2 x^{2}$

$4 x$

$1$

$x$

$2$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$9 x$

$5$

$3 x^{4}$

$6 x^{3}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$9 x$

$4 x^{2}$

$8 x$

$x$

$5$

$x$

$2$

$7$

Step 21

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

$3 x^{3} - 2 x^{2} - 4 x + 1 + \frac{7}{x - 2}$

Divide Using Long Polynomial Division (3x^4-8x^3+9x+5)÷(x-2)

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