Algebra Examples

$(8 x^{3} - 14 x^{2} + 19 x - 9) \div (1 - 4 x)$

Step 1

To calculate the remainder, first divide the polynomials.

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Step 1.1

Reorder $1$ and $- 4 x$ .

$\frac{8 x^{3} - 14 x^{2} + 19 x - 9}{- 4 x + 1}$

Step 1.2

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

$4 x$

$1$

$8 x^{3}$

$14 x^{2}$

$19 x$

$9$

Step 1.3

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

				-	$2 x^{2}$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$

Step 1.4

Multiply the new quotient term by the divisor.

				-	$2 x^{2}$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				+	$8 x^{3}$	-	$2 x^{2}$

Step 1.5

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{3} - 2 x^{2}$

				-	$2 x^{2}$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				-	$8 x^{3}$	+	$2 x^{2}$

Step 1.6

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

				-	$2 x^{2}$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				-	$8 x^{3}$	+	$2 x^{2}$
						-	$12 x^{2}$

Step 1.7

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

				-	$2 x^{2}$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				-	$8 x^{3}$	+	$2 x^{2}$
						-	$12 x^{2}$	+	$19 x$

Step 1.8

Divide the highest order term in the dividend $- 12 x^{2}$ by the highest order term in divisor $- 4 x$ .

				-	$2 x^{2}$	+	$3 x$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				-	$8 x^{3}$	+	$2 x^{2}$
						-	$12 x^{2}$	+	$19 x$

Step 1.9

Multiply the new quotient term by the divisor.

				-	$2 x^{2}$	+	$3 x$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				-	$8 x^{3}$	+	$2 x^{2}$
						-	$12 x^{2}$	+	$19 x$
						-	$12 x^{2}$	+	$3 x$

Step 1.10

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{2} + 3 x$

				-	$2 x^{2}$	+	$3 x$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				-	$8 x^{3}$	+	$2 x^{2}$
						-	$12 x^{2}$	+	$19 x$
						+	$12 x^{2}$	-	$3 x$

Step 1.11

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

				-	$2 x^{2}$	+	$3 x$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				-	$8 x^{3}$	+	$2 x^{2}$
						-	$12 x^{2}$	+	$19 x$
						+	$12 x^{2}$	-	$3 x$
								+	$16 x$

Step 1.12

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

				-	$2 x^{2}$	+	$3 x$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				-	$8 x^{3}$	+	$2 x^{2}$
						-	$12 x^{2}$	+	$19 x$
						+	$12 x^{2}$	-	$3 x$
								+	$16 x$	-	$9$

Step 1.13

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $- 4 x$ .

				-	$2 x^{2}$	+	$3 x$	-	$4$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				-	$8 x^{3}$	+	$2 x^{2}$
						-	$12 x^{2}$	+	$19 x$
						+	$12 x^{2}$	-	$3 x$
								+	$16 x$	-	$9$

Step 1.14

Multiply the new quotient term by the divisor.

				-	$2 x^{2}$	+	$3 x$	-	$4$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				-	$8 x^{3}$	+	$2 x^{2}$
						-	$12 x^{2}$	+	$19 x$
						+	$12 x^{2}$	-	$3 x$
								+	$16 x$	-	$9$
								+	$16 x$	-	$4$

Step 1.15

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

				-	$2 x^{2}$	+	$3 x$	-	$4$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				-	$8 x^{3}$	+	$2 x^{2}$
						-	$12 x^{2}$	+	$19 x$
						+	$12 x^{2}$	-	$3 x$
								+	$16 x$	-	$9$
								-	$16 x$	+	$4$

Step 1.16

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

				-	$2 x^{2}$	+	$3 x$	-	$4$
-	$4 x$	+	$1$		$8 x^{3}$	-	$14 x^{2}$	+	$19 x$	-	$9$
				-	$8 x^{3}$	+	$2 x^{2}$
						-	$12 x^{2}$	+	$19 x$
						+	$12 x^{2}$	-	$3 x$
								+	$16 x$	-	$9$
								-	$16 x$	+	$4$
										-	$5$

Step 1.17

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

$- 2 x^{2} + 3 x - 4 - \frac{5}{- 4 x + 1}$

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$- 5$