Divide Using Long Polynomial Division (5x^4+14x^3+9x)÷(x^2+3x+1)

$(5 x^{4} + 14 x^{3} + 9 x) \div (x^{2} + 3 x + 1)$

Step 1

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

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

Step 2

Divide the highest order term in the dividend $5 x^{4}$ by the highest order term in divisor $x^{2}$ .

$5 x^{2}$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$5 x^{2}$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{4} + 15 x^{3} + 5 x^{2}$

$5 x^{2}$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

Step 5

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

$5 x^{2}$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

$x^{3}$

$5 x^{2}$

Step 6

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

$5 x^{2}$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

$x^{3}$

$5 x^{2}$

$9 x$

Step 7

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

$5 x^{2}$

$x$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

$x^{3}$

$5 x^{2}$

$9 x$

Step 8

Multiply the new quotient term by the divisor.

$5 x^{2}$

$x$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

$x^{3}$

$5 x^{2}$

$9 x$

$x^{3}$

$3 x^{2}$

$x$

Step 9

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

$5 x^{2}$

$x$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

$x^{3}$

$5 x^{2}$

$9 x$

$x^{3}$

$3 x^{2}$

$x$

Step 10

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

$5 x^{2}$

$x$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

$x^{3}$

$5 x^{2}$

$9 x$

$x^{3}$

$3 x^{2}$

$x$

$2 x^{2}$

$10 x$

Step 11

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

$5 x^{2}$

$x$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

$x^{3}$

$5 x^{2}$

$9 x$

$x^{3}$

$3 x^{2}$

$x$

$2 x^{2}$

$10 x$

$0$

Step 12

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

$5 x^{2}$

$x$

$2$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

$x^{3}$

$5 x^{2}$

$9 x$

$x^{3}$

$3 x^{2}$

$x$

$2 x^{2}$

$10 x$

$0$

Step 13

Multiply the new quotient term by the divisor.

$5 x^{2}$

$x$

$2$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

$x^{3}$

$5 x^{2}$

$9 x$

$x^{3}$

$3 x^{2}$

$x$

$2 x^{2}$

$10 x$

$0$

$2 x^{2}$

$6 x$

$2$

Step 14

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

$5 x^{2}$

$x$

$2$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

$x^{3}$

$5 x^{2}$

$9 x$

$x^{3}$

$3 x^{2}$

$x$

$2 x^{2}$

$10 x$

$0$

$2 x^{2}$

$6 x$

$2$

Step 15

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

$5 x^{2}$

$x$

$2$

$x^{2}$

$3 x$

$1$

$5 x^{4}$

$14 x^{3}$

$0 x^{2}$

$9 x$

$0$

$5 x^{4}$

$15 x^{3}$

$5 x^{2}$

$x^{3}$

$5 x^{2}$

$9 x$

$x^{3}$

$3 x^{2}$

$x$

$2 x^{2}$

$10 x$

$0$

$2 x^{2}$

$6 x$

$2$

$16 x$

$2$

Step 16

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

$5 x^{2} - x - 2 + \frac{16 x + 2}{x^{2} + 3 x + 1}$

Divide Using Long Polynomial Division (5x^4+14x^3+9x)÷(x^2+3x+1)

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