Precalculus Examples

$\frac{4 x^{4} - 4 x^{2} + 6 x}{x - 4}$

Step 1

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

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

Step 2

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

$4 x^{3}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$4 x^{3}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

Step 4

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

$4 x^{3}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

Step 5

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

$4 x^{3}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

Step 6

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

$4 x^{3}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

Step 7

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

$4 x^{3}$

$16 x^{2}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$4 x^{3}$

$16 x^{2}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $16 x^{3} - 64 x^{2}$

$4 x^{3}$

$16 x^{2}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

Step 10

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

$4 x^{3}$

$16 x^{2}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

$60 x^{2}$

Step 11

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

$4 x^{3}$

$16 x^{2}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

$60 x^{2}$

$6 x$

Step 12

Divide the highest order term in the dividend $60 x^{2}$ by the highest order term in divisor $x$ .

$4 x^{3}$

$16 x^{2}$

$60 x$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

$60 x^{2}$

$6 x$

Step 13

Multiply the new quotient term by the divisor.

$4 x^{3}$

$16 x^{2}$

$60 x$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

$60 x^{2}$

$6 x$

$60 x^{2}$

$240 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $60 x^{2} - 240 x$

$4 x^{3}$

$16 x^{2}$

$60 x$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

$60 x^{2}$

$6 x$

$60 x^{2}$

$240 x$

Step 15

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

$4 x^{3}$

$16 x^{2}$

$60 x$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

$60 x^{2}$

$6 x$

$60 x^{2}$

$240 x$

$246 x$

Step 16

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

$4 x^{3}$

$16 x^{2}$

$60 x$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

$60 x^{2}$

$6 x$

$60 x^{2}$

$240 x$

$246 x$

$0$

Step 17

Divide the highest order term in the dividend $246 x$ by the highest order term in divisor $x$ .

$4 x^{3}$

$16 x^{2}$

$60 x$

$246$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

$60 x^{2}$

$6 x$

$60 x^{2}$

$240 x$

$246 x$

$0$

Step 18

Multiply the new quotient term by the divisor.

$4 x^{3}$

$16 x^{2}$

$60 x$

$246$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

$60 x^{2}$

$6 x$

$60 x^{2}$

$240 x$

$246 x$

$0$

$246 x$

$984$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $246 x - 984$

$4 x^{3}$

$16 x^{2}$

$60 x$

$246$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

$60 x^{2}$

$6 x$

$60 x^{2}$

$240 x$

$246 x$

$0$

$246 x$

$984$

Step 20

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

$4 x^{3}$

$16 x^{2}$

$60 x$

$246$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$4 x^{2}$

$6 x$

$0$

$4 x^{4}$

$16 x^{3}$

$4 x^{2}$

$16 x^{3}$

$64 x^{2}$

$60 x^{2}$

$6 x$

$60 x^{2}$

$240 x$

$246 x$

$0$

$246 x$

$984$

Step 21

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

$4 x^{3} + 16 x^{2} + 60 x + 246 + \frac{984}{x - 4}$