Pre-Algebra Examples

$\frac{- x^{3} + 23 x^{2} - 104 x - 308}{x - 11}$

Step 1

Write $\frac{- x^{3} + 23 x^{2} - 104 x - 308}{x - 11}$ as an equation.

$y = \frac{- x^{3} + 23 x^{2} - 104 x - 308}{x - 11}$

Step 2

Simplify $\frac{- x^{3} + 23 x^{2} - 104 x - 308}{x - 11}$ .

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Step 2.1

Split the fraction $\frac{- x^{3} + 23 x^{2} - 104 x - 308}{x - 11}$ into two fractions.

$y = \frac{- x^{3} + 23 x^{2} - 104 x}{x - 11} + \frac{- 308}{x - 11}$

Step 2.2

Simplify each term.

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Step 2.2.1

Factor $x$ out of $- x^{3} + 23 x^{2} - 104 x$ .

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Step 2.2.1.1

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

$y = \frac{x (- x^{2}) + 23 x^{2} - 104 x}{x - 11} + \frac{- 308}{x - 11}$

Step 2.2.1.2

Factor $x$ out of $23 x^{2}$ .

$y = \frac{x (- x^{2}) + x (23 x) - 104 x}{x - 11} + \frac{- 308}{x - 11}$

Step 2.2.1.3

Factor $x$ out of $- 104 x$ .

$y = \frac{x (- x^{2}) + x (23 x) + x \cdot - 104}{x - 11} + \frac{- 308}{x - 11}$

Step 2.2.1.4

Factor $x$ out of $x (- x^{2}) + x (23 x)$ .

$y = \frac{x (- x^{2} + 23 x) + x \cdot - 104}{x - 11} + \frac{- 308}{x - 11}$

Step 2.2.1.5

Factor $x$ out of $x (- x^{2} + 23 x) + x \cdot - 104$ .

$y = \frac{x (- x^{2} + 23 x - 104)}{x - 11} + \frac{- 308}{x - 11}$

Step 2.2.2

Move the negative in front of the fraction.

$y = \frac{x (- x^{2} + 23 x - 104)}{x - 11} - \frac{308}{x - 11}$

Step 3

Combine the numerators over the common denominator.

$y = \frac{x (- x^{2} + 23 x - 104) - 308}{x - 11}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$y = \frac{x (- x^{2}) + x (23 x) + x \cdot - 104 - 308}{x - 11}$

Step 4.2

Simplify.

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Step 4.2.1

Rewrite using the commutative property of multiplication.

$y = \frac{- x \cdot x^{2} + x (23 x) + x \cdot - 104 - 308}{x - 11}$

Step 4.2.2

Rewrite using the commutative property of multiplication.

$y = \frac{- x \cdot x^{2} + 23 x \cdot x + x \cdot - 104 - 308}{x - 11}$

Step 4.2.3

Move $- 104$ to the left of $x$ .

$y = \frac{- x \cdot x^{2} + 23 x \cdot x - 104 \cdot x - 308}{x - 11}$

Step 4.3

Simplify each term.

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Step 4.3.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.3.1.1

Move $x^{2}$ .

$y = \frac{- (x^{2} x) + 23 x \cdot x - 104 \cdot x - 308}{x - 11}$

Step 4.3.1.2

Multiply $x^{2}$ by $x$ .

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Step 4.3.1.2.1

Raise $x$ to the power of $1$ .

$y = \frac{- (x^{2} x^{1}) + 23 x \cdot x - 104 \cdot x - 308}{x - 11}$

Step 4.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{- x^{2 + 1} + 23 x \cdot x - 104 \cdot x - 308}{x - 11}$

Step 4.3.1.3

Add $2$ and $1$ .

$y = \frac{- x^{3} + 23 x \cdot x - 104 \cdot x - 308}{x - 11}$

Step 4.3.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.2.1

Move $x$ .

$y = \frac{- x^{3} + 23 (x \cdot x) - 104 \cdot x - 308}{x - 11}$

Step 4.3.2.2

Multiply $x$ by $x$ .

$y = \frac{- x^{3} + 23 x^{2} - 104 \cdot x - 308}{x - 11}$

$y = \frac{- x^{3} + 23 x^{2} - 104 x - 308}{x - 11}$

Step 4.4

Rewrite $- x^{3} + 23 x^{2} - 104 x - 308$ in a factored form.

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Step 4.4.1

Factor $- x^{3} + 23 x^{2} - 104 x - 308$ using the rational roots test.

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Step 4.4.1.1

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.

$p = \pm 1, \pm 308, \pm 2, \pm 154, \pm 4, \pm 77, \pm 7, \pm 44, \pm 11, \pm 28, \pm 14, \pm 22$

$q = \pm 1$

Step 4.4.1.2

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

$\pm 1, \pm 308, \pm 2, \pm 154, \pm 4, \pm 77, \pm 7, \pm 44, \pm 11, \pm 28, \pm 14, \pm 22$

Step 4.4.1.3

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

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Step 4.4.1.3.1

Substitute $- 2$ into the polynomial.

$- {(- 2)}^{3} + 23 {(- 2)}^{2} - 104 \cdot - 2 - 308$

Step 4.4.1.3.2

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

$- - 8 + 23 {(- 2)}^{2} - 104 \cdot - 2 - 308$

Step 4.4.1.3.3

Multiply $- 1$ by $- 8$ .

$8 + 23 {(- 2)}^{2} - 104 \cdot - 2 - 308$

Step 4.4.1.3.4

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

$8 + 23 \cdot 4 - 104 \cdot - 2 - 308$

Step 4.4.1.3.5

Multiply $23$ by $4$ .

$8 + 92 - 104 \cdot - 2 - 308$

Step 4.4.1.3.6

Add $8$ and $92$ .

$100 - 104 \cdot - 2 - 308$

Step 4.4.1.3.7

Multiply $- 104$ by $- 2$ .

$100 + 208 - 308$

Step 4.4.1.3.8

Add $100$ and $208$ .

$308 - 308$

Step 4.4.1.3.9

Subtract $308$ from $308$ .

$0$

Step 4.4.1.4

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- x^{3} + 23 x^{2} - 104 x - 308}{x + 2}$

Step 4.4.1.5

Divide $- x^{3} + 23 x^{2} - 104 x - 308$ by $x + 2$ .

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Step 4.4.1.5.1

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

$x$

$2$

$x^{3}$

$23 x^{2}$

$104 x$

$308$

Step 4.4.1.5.2

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

				-	$x^{2}$
	$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$

Step 4.4.1.5.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			-	$x^{3}$	-	$2 x^{2}$

Step 4.4.1.5.4

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

			-	$x^{2}$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			+	$x^{3}$	+	$2 x^{2}$

Step 4.4.1.5.5

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

			-	$x^{2}$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			+	$x^{3}$	+	$2 x^{2}$
					+	$25 x^{2}$

Step 4.4.1.5.6

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

			-	$x^{2}$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			+	$x^{3}$	+	$2 x^{2}$
					+	$25 x^{2}$	-	$104 x$

Step 4.4.1.5.7

Divide the highest order term in the dividend $25 x^{2}$ by the highest order term in divisor $x$ .

			-	$x^{2}$	+	$25 x$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			+	$x^{3}$	+	$2 x^{2}$
					+	$25 x^{2}$	-	$104 x$

Step 4.4.1.5.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$25 x$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			+	$x^{3}$	+	$2 x^{2}$
					+	$25 x^{2}$	-	$104 x$
					+	$25 x^{2}$	+	$50 x$

Step 4.4.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $25 x^{2} + 50 x$

			-	$x^{2}$	+	$25 x$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			+	$x^{3}$	+	$2 x^{2}$
					+	$25 x^{2}$	-	$104 x$
					-	$25 x^{2}$	-	$50 x$

Step 4.4.1.5.10

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

			-	$x^{2}$	+	$25 x$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			+	$x^{3}$	+	$2 x^{2}$
					+	$25 x^{2}$	-	$104 x$
					-	$25 x^{2}$	-	$50 x$
							-	$154 x$

Step 4.4.1.5.11

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

			-	$x^{2}$	+	$25 x$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			+	$x^{3}$	+	$2 x^{2}$
					+	$25 x^{2}$	-	$104 x$
					-	$25 x^{2}$	-	$50 x$
							-	$154 x$	-	$308$

Step 4.4.1.5.12

Divide the highest order term in the dividend $- 154 x$ by the highest order term in divisor $x$ .

			-	$x^{2}$	+	$25 x$	-	$154$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			+	$x^{3}$	+	$2 x^{2}$
					+	$25 x^{2}$	-	$104 x$
					-	$25 x^{2}$	-	$50 x$
							-	$154 x$	-	$308$

Step 4.4.1.5.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$25 x$	-	$154$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			+	$x^{3}$	+	$2 x^{2}$
					+	$25 x^{2}$	-	$104 x$
					-	$25 x^{2}$	-	$50 x$
							-	$154 x$	-	$308$
							-	$154 x$	-	$308$

Step 4.4.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 154 x - 308$

			-	$x^{2}$	+	$25 x$	-	$154$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			+	$x^{3}$	+	$2 x^{2}$
					+	$25 x^{2}$	-	$104 x$
					-	$25 x^{2}$	-	$50 x$
							-	$154 x$	-	$308$
							+	$154 x$	+	$308$

Step 4.4.1.5.15

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

			-	$x^{2}$	+	$25 x$	-	$154$
$x$	+	$2$	-	$x^{3}$	+	$23 x^{2}$	-	$104 x$	-	$308$
			+	$x^{3}$	+	$2 x^{2}$
					+	$25 x^{2}$	-	$104 x$
					-	$25 x^{2}$	-	$50 x$
							-	$154 x$	-	$308$
							+	$154 x$	+	$308$
										$0$

Step 4.4.1.5.16

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

$- x^{2} + 25 x - 154$

Step 4.4.1.6

Write $- x^{3} + 23 x^{2} - 104 x - 308$ as a set of factors.

$y = \frac{(x + 2) (- x^{2} + 25 x - 154)}{x - 11}$

Step 4.4.2

Factor by grouping.

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Step 4.4.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 154 = 154$ and whose sum is $b = 25$ .

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Step 4.4.2.1.1

Factor $25$ out of $25 x$ .

$y = \frac{(x + 2) (- x^{2} + 25 (x) - 154)}{x - 11}$

Step 4.4.2.1.2

Rewrite $25$ as $11$ plus $14$

$y = \frac{(x + 2) (- x^{2} + (11 + 14) x - 154)}{x - 11}$

Step 4.4.2.1.3

Apply the distributive property.

$y = \frac{(x + 2) (- x^{2} + 11 x + 14 x - 154)}{x - 11}$

Step 4.4.2.2

Factor out the greatest common factor from each group.

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Step 4.4.2.2.1

Group the first two terms and the last two terms.

$y = \frac{(x + 2) ((- x^{2} + 11 x) + 14 x - 154)}{x - 11}$

Step 4.4.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{(x + 2) (x (- x + 11) - 14 (- x + 11))}{x - 11}$

Step 4.4.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 11$ .

$y = \frac{(x + 2) ((- x + 11) (x - 14))}{x - 11}$

$y = \frac{(x + 2) (- x + 11) (x - 14)}{x - 11}$

Step 5

Cancel the common factor of $- x + 11$ and $x - 11$ .

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Step 5.1

Factor $- 1$ out of $- x$ .

$y = \frac{(x + 2) (- (x) + 11) (x - 14)}{x - 11}$

Step 5.2

Rewrite $11$ as $- 1 (- 11)$ .

$y = \frac{(x + 2) (- (x) - 1 (- 11)) (x - 14)}{x - 11}$

Step 5.3

Factor $- 1$ out of $- (x) - 1 (- 11)$ .

$y = \frac{(x + 2) (- (x - 11)) (x - 14)}{x - 11}$

Step 5.4

Rewrite $- (x - 11)$ as $- 1 (x - 11)$ .

$y = \frac{(x + 2) (- 1 (x - 11)) (x - 14)}{x - 11}$

Step 5.5

Cancel the common factor.

$y = \frac{(x + 2) (- 1 (x - 11)) (x - 14)}{x - 11}$

Step 5.6

Divide $((x + 2) \cdot (- 1)) (x - 14)$ by $1$ .

$y = ((x + 2) \cdot (- 1)) (x - 14)$

Step 6

Apply the distributive property.

$y = (x \cdot - 1 + 2 \cdot - 1) (x - 14)$

Step 7

Move $- 1$ to the left of $x$ .

$y = (- 1 \cdot x + 2 \cdot - 1) (x - 14)$

Step 8

Multiply $2$ by $- 1$ .

$y = (- 1 \cdot x - 2) (x - 14)$

Step 9

Rewrite $- 1 x$ as $- x$ .

$y = (- x - 2) (x - 14)$

Step 10

Expand $(- x - 2) (x - 14)$ using the FOIL Method.

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Step 10.1

Apply the distributive property.

$y = - x (x - 14) - 2 (x - 14)$

Step 10.2

Apply the distributive property.

$y = - x \cdot x - x \cdot - 14 - 2 (x - 14)$

Step 10.3

Apply the distributive property.

$y = - x \cdot x - x \cdot - 14 - 2 x - 2 \cdot - 14$

Step 11

Simplify and combine like terms.

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Step 11.1

Simplify each term.

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Step 11.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 11.1.1.1

Move $x$ .

$y = - (x \cdot x) - x \cdot - 14 - 2 x - 2 \cdot - 14$

Step 11.1.1.2

Multiply $x$ by $x$ .

$y = - x^{2} - x \cdot - 14 - 2 x - 2 \cdot - 14$

Step 11.1.2

Multiply $- 14$ by $- 1$ .

$y = - x^{2} + 14 x - 2 x - 2 \cdot - 14$

Step 11.1.3

Multiply $- 2$ by $- 14$ .

$y = - x^{2} + 14 x - 2 x + 28$

Step 11.2

Subtract $2 x$ from $14 x$ .

$y = - x^{2} + 12 x + 28$

Step 12

When solving for $y$ , $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x^{2}$