Algebra Examples

$\frac{x^{4} - 49 x^{2}}{x^{3} + 2 x^{4} - 63 x}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $x^{2}$ out of $x^{4} - 49 x^{2}$ .

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Step 1.1.1

Factor $x^{2}$ out of $x^{4}$ .

$\frac{x^{2} x^{2} - 49 x^{2}}{x^{3} + 2 x^{4} - 63 x}$

Step 1.1.2

Factor $x^{2}$ out of $- 49 x^{2}$ .

$\frac{x^{2} x^{2} + x^{2} \cdot - 49}{x^{3} + 2 x^{4} - 63 x}$

Step 1.1.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 49$ .

$\frac{x^{2} (x^{2} - 49)}{x^{3} + 2 x^{4} - 63 x}$

Step 1.2

Rewrite $49$ as $7^{2}$ .

$\frac{x^{2} (x^{2} - 7^{2})}{x^{3} + 2 x^{4} - 63 x}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$\frac{x^{2} (x + 7) (x - 7)}{x^{3} + 2 x^{4} - 63 x}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $x$ out of $x^{3} + 2 x^{4} - 63 x$ .

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Step 2.1.1

Factor $x$ out of $x^{3}$ .

$\frac{x^{2} (x + 7) (x - 7)}{x \cdot x^{2} + 2 x^{4} - 63 x}$

Step 2.1.2

Factor $x$ out of $2 x^{4}$ .

$\frac{x^{2} (x + 7) (x - 7)}{x \cdot x^{2} + x (2 x^{3}) - 63 x}$

Step 2.1.3

Factor $x$ out of $- 63 x$ .

$\frac{x^{2} (x + 7) (x - 7)}{x \cdot x^{2} + x (2 x^{3}) + x \cdot - 63}$

Step 2.1.4

Factor $x$ out of $x \cdot x^{2} + x (2 x^{3})$ .

$\frac{x^{2} (x + 7) (x - 7)}{x (x^{2} + 2 x^{3}) + x \cdot - 63}$

Step 2.1.5

Factor $x$ out of $x (x^{2} + 2 x^{3}) + x \cdot - 63$ .

$\frac{x^{2} (x + 7) (x - 7)}{x (x^{2} + 2 x^{3} - 63)}$

Step 2.2

Reorder terms.

$\frac{x^{2} (x + 7) (x - 7)}{x (2 x^{3} + x^{2} - 63)}$

Step 2.3

Factor $2 x^{3} + x^{2} - 63$ using the rational roots test.

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Step 2.3.1

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.

$p = \pm 1, \pm 63, \pm 3, \pm 21, \pm 7, \pm 9$

$q = \pm 1, \pm 2$

Step 2.3.2

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

$\pm 1, \pm 0.5, \pm 63, \pm 31.5, \pm 3, \pm 1.5, \pm 21, \pm 10.5, \pm 7, \pm 3.5, \pm 9, \pm 4.5$

Step 2.3.3

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 2.3.3.1

Substitute $3$ into the polynomial.

$2 \cdot 3^{3} + 3^{2} - 63$

Step 2.3.3.2

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

$2 \cdot 27 + 3^{2} - 63$

Step 2.3.3.3

Multiply $2$ by $27$ .

$54 + 3^{2} - 63$

Step 2.3.3.4

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

$54 + 9 - 63$

Step 2.3.3.5

Add $54$ and $9$ .

$63 - 63$

Step 2.3.3.6

Subtract $63$ from $63$ .

$0$

Step 2.3.4

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.

$\frac{2 x^{3} + x^{2} - 63}{x - 3}$

Step 2.3.5

Divide $2 x^{3} + x^{2} - 63$ by $x - 3$ .

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Step 2.3.5.1

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

$x$

$3$

$2 x^{3}$

$x^{2}$

$0 x$

$63$

Step 2.3.5.2

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

					$2 x^{2}$
	$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$

Step 2.3.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			+	$2 x^{3}$	-	$6 x^{2}$

Step 2.3.5.4

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

				$2 x^{2}$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			-	$2 x^{3}$	+	$6 x^{2}$

Step 2.3.5.5

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

				$2 x^{2}$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$7 x^{2}$

Step 2.3.5.6

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

				$2 x^{2}$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$7 x^{2}$	+	$0 x$

Step 2.3.5.7

Divide the highest order term in the dividend $7 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$7 x$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$7 x^{2}$	+	$0 x$

Step 2.3.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$7 x$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					+	$7 x^{2}$	-	$21 x$

Step 2.3.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} - 21 x$

				$2 x^{2}$	+	$7 x$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	+	$21 x$

Step 2.3.5.10

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

				$2 x^{2}$	+	$7 x$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	+	$21 x$
							+	$21 x$

Step 2.3.5.11

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

				$2 x^{2}$	+	$7 x$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	+	$21 x$
							+	$21 x$	-	$63$

Step 2.3.5.12

Divide the highest order term in the dividend $21 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$7 x$	+	$21$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	+	$21 x$
							+	$21 x$	-	$63$

Step 2.3.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$7 x$	+	$21$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	+	$21 x$
							+	$21 x$	-	$63$
							+	$21 x$	-	$63$

Step 2.3.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $21 x - 63$

				$2 x^{2}$	+	$7 x$	+	$21$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	+	$21 x$
							+	$21 x$	-	$63$
							-	$21 x$	+	$63$

Step 2.3.5.15

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

				$2 x^{2}$	+	$7 x$	+	$21$
$x$	-	$3$		$2 x^{3}$	+	$x^{2}$	+	$0 x$	-	$63$
			-	$2 x^{3}$	+	$6 x^{2}$
					+	$7 x^{2}$	+	$0 x$
					-	$7 x^{2}$	+	$21 x$
							+	$21 x$	-	$63$
							-	$21 x$	+	$63$
										$0$

Step 2.3.5.16

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

$2 x^{2} + 7 x + 21$

Step 2.3.6

Write $2 x^{3} + x^{2} - 63$ as a set of factors.

$\frac{x^{2} (x + 7) (x - 7)}{x ((x - 3) (2 x^{2} + 7 x + 21))}$

$\frac{x^{2} (x + 7) (x - 7)}{x (x - 3) (2 x^{2} + 7 x + 21)}$

Step 3

Cancel the common factor of $x^{2}$ and $x$ .

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Step 3.1

Factor $x$ out of $x^{2} (x + 7) (x - 7)$ .

$\frac{x ((x (x + 7)) (x - 7))}{x (x - 3) (2 x^{2} + 7 x + 21)}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x$ out of $x (x - 3) (2 x^{2} + 7 x + 21)$ .

$\frac{x ((x (x + 7)) (x - 7))}{x ((x - 3) (2 x^{2} + 7 x + 21))}$

Step 3.2.2

Cancel the common factor.

$\frac{x ((x (x + 7)) (x - 7))}{x ((x - 3) (2 x^{2} + 7 x + 21))}$

Step 3.2.3

Rewrite the expression.

$\frac{(x (x + 7)) (x - 7)}{(x - 3) (2 x^{2} + 7 x + 21)}$

$\frac{x (x + 7) (x - 7)}{(x - 3) (2 x^{2} + 7 x + 21)}$