Divide Using Long Polynomial Division (5x^4-3x^2-4x+6)÷(x-7)

$(5 x^{4} - 3 x^{2} - 4 x + 6) \div (x - 7)$

Step 1

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

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

Step 2

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

$5 x^{3}$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

Step 3

Multiply the new quotient term by the divisor.

$5 x^{3}$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{4} - 35 x^{3}$

$5 x^{3}$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

Step 5

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

$5 x^{3}$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

Step 6

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

$5 x^{3}$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

Step 7

Divide the highest order term in the dividend $35 x^{3}$ by the highest order term in divisor $x$ .

$5 x^{3}$

$35 x^{2}$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$5 x^{3}$

$35 x^{2}$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $35 x^{3} - 245 x^{2}$

$5 x^{3}$

$35 x^{2}$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

Step 10

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

$5 x^{3}$

$35 x^{2}$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

$242 x^{2}$

Step 11

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

$5 x^{3}$

$35 x^{2}$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

$242 x^{2}$

$4 x$

Step 12

Divide the highest order term in the dividend $242 x^{2}$ by the highest order term in divisor $x$ .

$5 x^{3}$

$35 x^{2}$

$242 x$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

$242 x^{2}$

$4 x$

Step 13

Multiply the new quotient term by the divisor.

$5 x^{3}$

$35 x^{2}$

$242 x$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

$242 x^{2}$

$4 x$

$242 x^{2}$

$1694 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $242 x^{2} - 1694 x$

$5 x^{3}$

$35 x^{2}$

$242 x$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

$242 x^{2}$

$4 x$

$242 x^{2}$

$1694 x$

Step 15

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

$5 x^{3}$

$35 x^{2}$

$242 x$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

$242 x^{2}$

$4 x$

$242 x^{2}$

$1694 x$

$1690 x$

Step 16

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

$5 x^{3}$

$35 x^{2}$

$242 x$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

$242 x^{2}$

$4 x$

$242 x^{2}$

$1694 x$

$1690 x$

$6$

Step 17

Divide the highest order term in the dividend $1690 x$ by the highest order term in divisor $x$ .

$5 x^{3}$

$35 x^{2}$

$242 x$

$1690$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

$242 x^{2}$

$4 x$

$242 x^{2}$

$1694 x$

$1690 x$

$6$

Step 18

Multiply the new quotient term by the divisor.

$5 x^{3}$

$35 x^{2}$

$242 x$

$1690$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

$242 x^{2}$

$4 x$

$242 x^{2}$

$1694 x$

$1690 x$

$6$

$1690 x$

$11830$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $1690 x - 11830$

$5 x^{3}$

$35 x^{2}$

$242 x$

$1690$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

$242 x^{2}$

$4 x$

$242 x^{2}$

$1694 x$

$1690 x$

$6$

$1690 x$

$11830$

Step 20

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

$5 x^{3}$

$35 x^{2}$

$242 x$

$1690$

$x$

$7$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$4 x$

$6$

$5 x^{4}$

$35 x^{3}$

$3 x^{2}$

$35 x^{3}$

$245 x^{2}$

$242 x^{2}$

$4 x$

$242 x^{2}$

$1694 x$

$1690 x$

$6$

$1690 x$

$11830$

$11836$

Step 21

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

$5 x^{3} + 35 x^{2} + 242 x + 1690 + \frac{11836}{x - 7}$

Divide Using Long Polynomial Division (5x^4-3x^2-4x+6)÷(x-7)

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