Precalculus Examples

$\frac{3 x^{4} + 2 x^{2}}{x^{2} + 8 x^{4} + 6}$

Step 1

Reorder $x^{2}$ and $8 x^{4}$ .

$\frac{3 x^{4} + 2 x^{2}}{8 x^{4} + x^{2} + 6}$

Step 2

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

$8 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$3 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

Step 3

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $8 x^{4}$ .

$\frac{3}{8}$

$8 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$3 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

Step 4

Multiply the new quotient term by the divisor.

$\frac{3}{8}$

$8 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$3 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

$3 x^{4}$

$0$

$\frac{3 x^{2}}{8}$

$0$

$\frac{9}{4}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} + 0 + \frac{3 x^{2}}{8} + 0 + \frac{9}{4}$

$\frac{3}{8}$

$8 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$3 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

$3 x^{4}$

$0$

$\frac{3 x^{2}}{8}$

$0$

$\frac{9}{4}$

Step 6

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

$\frac{3}{8}$

$8 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$3 x^{4}$

$0 x^{3}$

$2 x^{2}$

$0 x$

$0$

$3 x^{4}$

$0$

$\frac{3 x^{2}}{8}$

$0$

$\frac{9}{4}$

$\frac{13 x^{2}}{8}$

$0$

$\frac{9}{4}$

Step 7

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

$\frac{3}{8} + \frac{\frac{13 x^{2}}{8} - \frac{9}{4}}{8 x^{4} + x^{2} + 6}$