Algebra Examples

$\frac{3 x^{3} + 4 x + 11}{x^{2} - 3 x + 2}$

Step 1

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

$x^{2}$

$3 x$

$2$

$3 x^{3}$

$0 x^{2}$

$4 x$

$11$

Step 2

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

$3 x$

$x^{2}$

$3 x$

$2$

$3 x^{3}$

$0 x^{2}$

$4 x$

$11$

Step 3

Multiply the new quotient term by the divisor.

$3 x$

$x^{2}$

$3 x$

$2$

$3 x^{3}$

$0 x^{2}$

$4 x$

$11$

$3 x^{3}$

$9 x^{2}$

$6 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} - 9 x^{2} + 6 x$

$3 x$

$x^{2}$

$3 x$

$2$

$3 x^{3}$

$0 x^{2}$

$4 x$

$11$

$3 x^{3}$

$9 x^{2}$

$6 x$

Step 5

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

$3 x$

$x^{2}$

$3 x$

$2$

$3 x^{3}$

$0 x^{2}$

$4 x$

$11$

$3 x^{3}$

$9 x^{2}$

$6 x$

$9 x^{2}$

$2 x$

Step 6

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

$3 x$

$x^{2}$

$3 x$

$2$

$3 x^{3}$

$0 x^{2}$

$4 x$

$11$

$3 x^{3}$

$9 x^{2}$

$6 x$

$9 x^{2}$

$2 x$

$11$

Step 7

Divide the highest order term in the dividend $9 x^{2}$ by the highest order term in divisor $x^{2}$ .

$3 x$

$9$

$x^{2}$

$3 x$

$2$

$3 x^{3}$

$0 x^{2}$

$4 x$

$11$

$3 x^{3}$

$9 x^{2}$

$6 x$

$9 x^{2}$

$2 x$

$11$

Step 8

Multiply the new quotient term by the divisor.

$3 x$

$9$

$x^{2}$

$3 x$

$2$

$3 x^{3}$

$0 x^{2}$

$4 x$

$11$

$3 x^{3}$

$9 x^{2}$

$6 x$

$9 x^{2}$

$2 x$

$11$

$9 x^{2}$

$27 x$

$18$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{2} - 27 x + 18$

$3 x$

$9$

$x^{2}$

$3 x$

$2$

$3 x^{3}$

$0 x^{2}$

$4 x$

$11$

$3 x^{3}$

$9 x^{2}$

$6 x$

$9 x^{2}$

$2 x$

$11$

$9 x^{2}$

$27 x$

$18$

Step 10

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

$3 x$

$9$

$x^{2}$

$3 x$

$2$

$3 x^{3}$

$0 x^{2}$

$4 x$

$11$

$3 x^{3}$

$9 x^{2}$

$6 x$

$9 x^{2}$

$2 x$

$11$

$9 x^{2}$

$27 x$

$18$

$25 x$

$7$

Step 11

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

$3 x + 9 + \frac{25 x - 7}{x^{2} - 3 x + 2}$