Precalculus Examples

$\frac{x^{6} + 17 x^{4} + 78 x^{2} + 54}{x^{2} + 9}$

Step 1

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

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

Step 2

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

$x^{4}$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

Step 3

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{6} + 0 + 9 x^{4}$

$x^{4}$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

Step 5

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

$x^{4}$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

$8 x^{4}$

Step 6

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

$x^{4}$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

$8 x^{4}$

$0 x^{3}$

$78 x^{2}$

Step 7

Divide the highest order term in the dividend $8 x^{4}$ by the highest order term in divisor $x^{2}$ .

$x^{4}$

$0 x^{3}$

$8 x^{2}$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

$8 x^{4}$

$0 x^{3}$

$78 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$x^{4}$

$0 x^{3}$

$8 x^{2}$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

$8 x^{4}$

$0 x^{3}$

$78 x^{2}$

$8 x^{4}$

$0$

$72 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{4} + 0 + 72 x^{2}$

$x^{4}$

$0 x^{3}$

$8 x^{2}$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

$8 x^{4}$

$0 x^{3}$

$78 x^{2}$

$8 x^{4}$

$0$

$72 x^{2}$

Step 10

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

$x^{4}$

$0 x^{3}$

$8 x^{2}$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

$8 x^{4}$

$0 x^{3}$

$78 x^{2}$

$8 x^{4}$

$0$

$72 x^{2}$

$6 x^{2}$

Step 11

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

$x^{4}$

$0 x^{3}$

$8 x^{2}$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

$8 x^{4}$

$0 x^{3}$

$78 x^{2}$

$8 x^{4}$

$0$

$72 x^{2}$

$6 x^{2}$

$0 x$

$54$

Step 12

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

$x^{4}$

$0 x^{3}$

$8 x^{2}$

$0 x$

$6$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

$8 x^{4}$

$0 x^{3}$

$78 x^{2}$

$8 x^{4}$

$0$

$72 x^{2}$

$6 x^{2}$

$0 x$

$54$

Step 13

Multiply the new quotient term by the divisor.

$x^{4}$

$0 x^{3}$

$8 x^{2}$

$0 x$

$6$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

$8 x^{4}$

$0 x^{3}$

$78 x^{2}$

$8 x^{4}$

$0$

$72 x^{2}$

$6 x^{2}$

$0 x$

$54$

$6 x^{2}$

$0$

$54$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} + 0 + 54$

$x^{4}$

$0 x^{3}$

$8 x^{2}$

$0 x$

$6$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

$8 x^{4}$

$0 x^{3}$

$78 x^{2}$

$8 x^{4}$

$0$

$72 x^{2}$

$6 x^{2}$

$0 x$

$54$

$6 x^{2}$

$0$

$54$

Step 15

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

$x^{4}$

$0 x^{3}$

$8 x^{2}$

$0 x$

$6$

$x^{2}$

$0 x$

$9$

$x^{6}$

$0 x^{5}$

$17 x^{4}$

$0 x^{3}$

$78 x^{2}$

$0 x$

$54$

$x^{6}$

$0$

$9 x^{4}$

$8 x^{4}$

$0 x^{3}$

$78 x^{2}$

$8 x^{4}$

$0$

$72 x^{2}$

$6 x^{2}$

$0 x$

$54$

$6 x^{2}$

$0$

$54$

$0$

Step 16

Since the remander is $0$ , the final answer is the quotient.

$x^{4} + 0 x^{3} + 8 x^{2} + 0 x + 6$