Algebra Examples

$\frac{x^{4} + 0 x^{3} - 10 x^{2} - 2 x + 3}{x + 3}$

Step 1

Simplify the numerator.

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Step 1.1

Multiply $0$ by $x^{3}$ .

$\frac{x^{4} + 0 - 10 x^{2} - 2 x + 3}{x + 3}$

Step 1.2

Add $x^{4}$ and $0$ .

$\frac{x^{4} - 10 x^{2} - 2 x + 3}{x + 3}$

Step 1.3

Factor $x^{4} - 10 x^{2} - 2 x + 3$ using the rational roots test.

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Step 1.3.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 3$

$q = \pm 1$

Step 1.3.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 3$

Step 1.3.3

Substitute $- 3$ and simplify the expression. In this case, the expression is equal to $0$ so $- 3$ is a root of the polynomial.

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Step 1.3.3.1

Substitute $- 3$ into the polynomial.

${(- 3)}^{4} - 10 {(- 3)}^{2} - 2 \cdot - 3 + 3$

Step 1.3.3.2

Raise $- 3$ to the power of $4$ .

$81 - 10 {(- 3)}^{2} - 2 \cdot - 3 + 3$

Step 1.3.3.3

Raise $- 3$ to the power of $2$ .

$81 - 10 \cdot 9 - 2 \cdot - 3 + 3$

Step 1.3.3.4

Multiply $- 10$ by $9$ .

$81 - 90 - 2 \cdot - 3 + 3$

Step 1.3.3.5

Subtract $90$ from $81$ .

$- 9 - 2 \cdot - 3 + 3$

Step 1.3.3.6

Multiply $- 2$ by $- 3$ .

$- 9 + 6 + 3$

Step 1.3.3.7

Add $- 9$ and $6$ .

$- 3 + 3$

Step 1.3.3.8

Add $- 3$ and $3$ .

$0$

Step 1.3.4

Since $- 3$ is a known root, divide the polynomial by $x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 10 x^{2} - 2 x + 3}{x + 3}$

Step 1.3.5

Divide $x^{4} - 10 x^{2} - 2 x + 3$ by $x + 3$ .

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Step 1.3.5.1

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

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

Step 1.3.5.2

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

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

Step 1.3.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

Step 1.3.5.4

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

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

Step 1.3.5.5

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

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

Step 1.3.5.6

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

$x^{3}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

Step 1.3.5.7

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

Step 1.3.5.8

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$3 x^{2}$
$x$	+	$3$		$x^{4}$	+	$0 x^{3}$	-	$10 x^{2}$	-	$2 x$	+	$3$
			-	$x^{4}$	-	$3 x^{3}$
					-	$3 x^{3}$	-	$10 x^{2}$
					-	$3 x^{3}$	-	$9 x^{2}$

Step 1.3.5.9

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$9 x^{2}$

Step 1.3.5.10

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$9 x^{2}$

$x^{2}$

Step 1.3.5.11

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$9 x^{2}$

$x^{2}$

$2 x$

Step 1.3.5.12

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$9 x^{2}$

$x^{2}$

$2 x$

Step 1.3.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$9 x^{2}$

$x^{2}$

$2 x$

$x^{2}$

$3 x$

Step 1.3.5.14

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$9 x^{2}$

$x^{2}$

$2 x$

$x^{2}$

$3 x$

Step 1.3.5.15

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$9 x^{2}$

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$x$

Step 1.3.5.16

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

$x^{3}$

$3 x^{2}$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$9 x^{2}$

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$x$

$3$

Step 1.3.5.17

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$9 x^{2}$

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$x$

$3$

Step 1.3.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$x$

$1$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$9 x^{2}$

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$x$

$3$

$x$

$3$

Step 1.3.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $x + 3$

$x^{3}$

$3 x^{2}$

$x$

$1$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$9 x^{2}$

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$x$

$3$

$x$

$3$

Step 1.3.5.20

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x$

$3$

$x^{4}$

$0 x^{3}$

$10 x^{2}$

$2 x$

$3$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$9 x^{2}$

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$x$

$3$

$x$

$3$

$0$

Step 1.3.5.21

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

$x^{3} - 3 x^{2} - x + 1$

Step 1.3.6

Write $x^{4} - 10 x^{2} - 2 x + 3$ as a set of factors.

$\frac{(x + 3) (x^{3} - 3 x^{2} - x + 1)}{x + 3}$

Step 2

Cancel the common factor of $x + 3$ .

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Step 2.1

Cancel the common factor.

$\frac{(x + 3) (x^{3} - 3 x^{2} - x + 1)}{x + 3}$

Step 2.2

Divide $x^{3} - 3 x^{2} - x + 1$ by $1$ .

$x^{3} - 3 x^{2} - x + 1$