Calculus Examples

$\frac{d y}{d x} + \frac{1}{x - 5} y = {(x - 5)}^{2}$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{1}{x - 5}$ .

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Step 1.1

Set up the integration.

$e^{\int \frac{1}{x - 5} d x}$

Step 1.2

Integrate $\frac{1}{x - 5}$ .

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Step 1.2.1

Let $u = x - 5$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.2.1.1

Let $u = x - 5$ . Find $\frac{d u}{d x}$ .

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Step 1.2.1.1.1

Differentiate $x - 5$ .

$\frac{d}{d x} [x - 5]$

Step 1.2.1.1.2

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$

Step 1.2.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 5]$

Step 1.2.1.1.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.2.1.1.5

Add $1$ and $0$ .

$1$

Step 1.2.1.2

Rewrite the problem using $u$ and $d u$ .

$e^{\int \frac{1}{u} d u}$

Step 1.2.2

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$e^{\ln (| u |) + C}$

Step 1.2.3

Replace all occurrences of $u$ with $x - 5$ .

$e^{\ln (| x - 5 |) + C}$

Step 1.3

Remove the constant of integration.

$e^{\ln (x - 5)}$

Step 1.4

Exponentiation and log are inverse functions.

$x - 5$

Step 2

Multiply each term by the integrating factor $x - 5$ .

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Step 2.1

Multiply each term by $x - 5$ .

$(x - 5) \frac{d y}{d x} + (x - 5) (\frac{1}{x - 5} y) = (x - 5) {(x - 5)}^{2}$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the distributive property.

$x \frac{d y}{d x} - 5 \frac{d y}{d x} + (x - 5) (\frac{1}{x - 5} y) = (x - 5) {(x - 5)}^{2}$

Step 2.2.2

Combine $\frac{1}{x - 5}$ and $y$ .

$x \frac{d y}{d x} - 5 \frac{d y}{d x} + (x - 5) \frac{y}{x - 5} = (x - 5) {(x - 5)}^{2}$

Step 2.2.3

Cancel the common factor of $x - 5$ .

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Step 2.2.3.1

Cancel the common factor.

$x \frac{d y}{d x} - 5 \frac{d y}{d x} + (x - 5) \frac{y}{x - 5} = (x - 5) {(x - 5)}^{2}$

Step 2.2.3.2

Rewrite the expression.

$x \frac{d y}{d x} - 5 \frac{d y}{d x} + y = (x - 5) {(x - 5)}^{2}$

Step 2.3

Multiply $x - 5$ by ${(x - 5)}^{2}$ by adding the exponents.

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Step 2.3.1

Multiply $x - 5$ by ${(x - 5)}^{2}$ .

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Step 2.3.1.1

Raise $x - 5$ to the power of $1$ .

$x \frac{d y}{d x} - 5 \frac{d y}{d x} + y = {(x - 5)}^{1} {(x - 5)}^{2}$

Step 2.3.1.2

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

$x \frac{d y}{d x} - 5 \frac{d y}{d x} + y = {(x - 5)}^{1 + 2}$

Step 2.3.2

Add $1$ and $2$ .

$x \frac{d y}{d x} - 5 \frac{d y}{d x} + y = {(x - 5)}^{3}$

Step 2.4

Use the Binomial Theorem.

$x \frac{d y}{d x} - 5 \frac{d y}{d x} + y = x^{3} + 3 x^{2} \cdot - 5 + 3 x {(- 5)}^{2} + {(- 5)}^{3}$

Step 2.5

Simplify each term.

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Step 2.5.1

Multiply $- 5$ by $3$ .

$x \frac{d y}{d x} - 5 \frac{d y}{d x} + y = x^{3} - 15 x^{2} + 3 x {(- 5)}^{2} + {(- 5)}^{3}$

Step 2.5.2

Raise $- 5$ to the power of $2$ .

$x \frac{d y}{d x} - 5 \frac{d y}{d x} + y = x^{3} - 15 x^{2} + 3 x \cdot 25 + {(- 5)}^{3}$

Step 2.5.3

Multiply $25$ by $3$ .

$x \frac{d y}{d x} - 5 \frac{d y}{d x} + y = x^{3} - 15 x^{2} + 75 x + {(- 5)}^{3}$

Step 2.5.4

Raise $- 5$ to the power of $3$ .

$x \frac{d y}{d x} - 5 \frac{d y}{d x} + y = x^{3} - 15 x^{2} + 75 x - 125$

Step 3

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [(x - 5) y] = x^{3} - 15 x^{2} + 75 x - 125$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [(x - 5) y] d x = \int x^{3} - 15 x^{2} + 75 x - 125 d x$

Step 5

Integrate the left side.

$(x - 5) y = \int x^{3} - 15 x^{2} + 75 x - 125 d x$

Step 6

Integrate the right side.

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Step 6.1

Split the single integral into multiple integrals.

$(x - 5) y = \int x^{3} d x + \int - 15 x^{2} d x + \int 75 x d x + \int - 125 d x$

Step 6.2

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$(x - 5) y = \frac{1}{4} x^{4} + C + \int - 15 x^{2} d x + \int 75 x d x + \int - 125 d x$

Step 6.3

Since $- 15$ is constant with respect to $x$ , move $- 15$ out of the integral.

$(x - 5) y = \frac{1}{4} x^{4} + C - 15 \int x^{2} d x + \int 75 x d x + \int - 125 d x$

Step 6.4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$(x - 5) y = \frac{1}{4} x^{4} + C - 15 (\frac{1}{3} x^{3} + C) + \int 75 x d x + \int - 125 d x$

Step 6.5

Since $75$ is constant with respect to $x$ , move $75$ out of the integral.

$(x - 5) y = \frac{1}{4} x^{4} + C - 15 (\frac{1}{3} x^{3} + C) + 75 \int x d x + \int - 125 d x$

Step 6.6

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$(x - 5) y = \frac{1}{4} x^{4} + C - 15 (\frac{1}{3} x^{3} + C) + 75 (\frac{1}{2} x^{2} + C) + \int - 125 d x$

Step 6.7

Apply the constant rule.

$(x - 5) y = \frac{1}{4} x^{4} + C - 15 (\frac{1}{3} x^{3} + C) + 75 (\frac{1}{2} x^{2} + C) - 125 x + C$

Step 6.8

Simplify.

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Step 6.8.1

Simplify.

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Step 6.8.1.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$(x - 5) y = \frac{1}{4} x^{4} + C - 15 (\frac{x^{3}}{3} + C) + 75 (\frac{1}{2} x^{2} + C) - 125 x + C$

Step 6.8.1.2

Combine $\frac{1}{2}$ and $x^{2}$ .

$(x - 5) y = \frac{1}{4} x^{4} + C - 15 (\frac{x^{3}}{3} + C) + 75 (\frac{x^{2}}{2} + C) - 125 x + C$

Step 6.8.2

Simplify.

$(x - 5) y = \frac{x^{4}}{4} - 5 x^{3} + \frac{75 x^{2}}{2} - 125 x + C$

Step 6.8.3

Reorder terms.

$(x - 5) y = \frac{1}{4} x^{4} - 5 x^{3} + \frac{75}{2} x^{2} - 125 x + C$

Step 7

Divide each term in $(x - 5) y = \frac{1}{4} x^{4} - 5 x^{3} + \frac{75}{2} x^{2} - 125 x + C$ by $x - 5$ and simplify.

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Step 7.1

Divide each term in $(x - 5) y = \frac{1}{4} x^{4} - 5 x^{3} + \frac{75}{2} x^{2} - 125 x + C$ by $x - 5$ .

$\frac{(x - 5) y}{x - 5} = \frac{\frac{1}{4} x^{4}}{x - 5} + \frac{- 5 x^{3}}{x - 5} + \frac{\frac{75}{2} x^{2}}{x - 5} + \frac{- 125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $x - 5$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{(x - 5) y}{x - 5} = \frac{\frac{1}{4} x^{4}}{x - 5} + \frac{- 5 x^{3}}{x - 5} + \frac{\frac{75}{2} x^{2}}{x - 5} + \frac{- 125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{4} x^{4}}{x - 5} + \frac{- 5 x^{3}}{x - 5} + \frac{\frac{75}{2} x^{2}}{x - 5} + \frac{- 125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$y = \frac{\frac{x^{4}}{4}}{x - 5} + \frac{- 5 x^{3}}{x - 5} + \frac{\frac{75}{2} x^{2}}{x - 5} + \frac{- 125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{x^{4}}{4} \cdot \frac{1}{x - 5} + \frac{- 5 x^{3}}{x - 5} + \frac{\frac{75}{2} x^{2}}{x - 5} + \frac{- 125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.1.3

Multiply $\frac{x^{4}}{4}$ by $\frac{1}{x - 5}$ .

$y = \frac{x^{4}}{4 (x - 5)} + \frac{- 5 x^{3}}{x - 5} + \frac{\frac{75}{2} x^{2}}{x - 5} + \frac{- 125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.1.4

Move the negative in front of the fraction.

$y = \frac{x^{4}}{4 (x - 5)} - \frac{5 x^{3}}{x - 5} + \frac{\frac{75}{2} x^{2}}{x - 5} + \frac{- 125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.1.5

Combine $\frac{75}{2}$ and $x^{2}$ .

$y = \frac{x^{4}}{4 (x - 5)} - \frac{5 x^{3}}{x - 5} + \frac{\frac{75 x^{2}}{2}}{x - 5} + \frac{- 125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.1.6

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{x^{4}}{4 (x - 5)} - \frac{5 x^{3}}{x - 5} + \frac{75 x^{2}}{2} \cdot \frac{1}{x - 5} + \frac{- 125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.1.7

Multiply $\frac{75 x^{2}}{2}$ by $\frac{1}{x - 5}$ .

$y = \frac{x^{4}}{4 (x - 5)} - \frac{5 x^{3}}{x - 5} + \frac{75 x^{2}}{2 (x - 5)} + \frac{- 125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.1.8

Move the negative in front of the fraction.

$y = \frac{x^{4}}{4 (x - 5)} - \frac{5 x^{3}}{x - 5} + \frac{75 x^{2}}{2 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.2

To write $- \frac{5 x^{3}}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{x^{4}}{4 (x - 5)} - \frac{5 x^{3}}{x - 5} \cdot \frac{4}{4} + \frac{75 x^{2}}{2 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.3

Write each expression with a common denominator of $4 (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.3.1

Multiply $\frac{5 x^{3}}{x - 5}$ by $\frac{4}{4}$ .

$y = \frac{x^{4}}{4 (x - 5)} - \frac{5 x^{3} \cdot 4}{(x - 5) \cdot 4} + \frac{75 x^{2}}{2 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.3.2

Reorder the factors of $(x - 5) \cdot 4$ .

$y = \frac{x^{4}}{4 (x - 5)} - \frac{5 x^{3} \cdot 4}{4 (x - 5)} + \frac{75 x^{2}}{2 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.4

Combine the numerators over the common denominator.

$y = \frac{x^{4} - 5 x^{3} \cdot 4}{4 (x - 5)} + \frac{75 x^{2}}{2 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.5

Simplify the numerator.

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Step 7.3.5.1

Factor $x^{3}$ out of $x^{4} - 5 x^{3} \cdot 4$ .

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Step 7.3.5.1.1

Factor $x^{3}$ out of $x^{4}$ .

$y = \frac{x^{3} x - 5 x^{3} \cdot 4}{4 (x - 5)} + \frac{75 x^{2}}{2 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.5.1.2

Factor $x^{3}$ out of $- 5 x^{3} \cdot 4$ .

$y = \frac{x^{3} x + x^{3} (- 5 \cdot 4)}{4 (x - 5)} + \frac{75 x^{2}}{2 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.5.1.3

Factor $x^{3}$ out of $x^{3} x + x^{3} (- 5 \cdot 4)$ .

$y = \frac{x^{3} (x - 5 \cdot 4)}{4 (x - 5)} + \frac{75 x^{2}}{2 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.5.2

Multiply $- 5$ by $4$ .

$y = \frac{x^{3} (x - 20)}{4 (x - 5)} + \frac{75 x^{2}}{2 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.6

To write $\frac{75 x^{2}}{2 (x - 5)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{x^{3} (x - 20)}{4 (x - 5)} + \frac{75 x^{2}}{2 (x - 5)} \cdot \frac{2}{2} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.7

Write each expression with a common denominator of $4 (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.7.1

Multiply $\frac{75 x^{2}}{2 (x - 5)}$ by $\frac{2}{2}$ .

$y = \frac{x^{3} (x - 20)}{4 (x - 5)} + \frac{75 x^{2} \cdot 2}{2 (x - 5) \cdot 2} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.7.2

Multiply $2$ by $2$ .

$y = \frac{x^{3} (x - 20)}{4 (x - 5)} + \frac{75 x^{2} \cdot 2}{4 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.8

Combine the numerators over the common denominator.

$y = \frac{x^{3} (x - 20) + 75 x^{2} \cdot 2}{4 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.9

Simplify the numerator.

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Step 7.3.9.1

Factor $x^{2}$ out of $x^{3} (x - 20) + 75 x^{2} \cdot 2$ .

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Step 7.3.9.1.1

Factor $x^{2}$ out of $x^{3} (x - 20)$ .

$y = \frac{x^{2} (x (x - 20)) + 75 x^{2} \cdot 2}{4 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.9.1.2

Factor $x^{2}$ out of $75 x^{2} \cdot 2$ .

$y = \frac{x^{2} (x (x - 20)) + x^{2} (75 \cdot 2)}{4 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.9.1.3

Factor $x^{2}$ out of $x^{2} (x (x - 20)) + x^{2} (75 \cdot 2)$ .

$y = \frac{x^{2} (x (x - 20) + 75 \cdot 2)}{4 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.9.2

Apply the distributive property.

$y = \frac{x^{2} (x \cdot x + x \cdot - 20 + 75 \cdot 2)}{4 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.9.3

Multiply $x$ by $x$ .

$y = \frac{x^{2} (x^{2} + x \cdot - 20 + 75 \cdot 2)}{4 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.9.4

Move $- 20$ to the left of $x$ .

$y = \frac{x^{2} (x^{2} - 20 \cdot x + 75 \cdot 2)}{4 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.9.5

Multiply $75$ by $2$ .

$y = \frac{x^{2} (x^{2} - 20 x + 150)}{4 (x - 5)} - \frac{125 x}{x - 5} + \frac{C}{x - 5}$

Step 7.3.10

To write $- \frac{125 x}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{x^{2} (x^{2} - 20 x + 150)}{4 (x - 5)} - \frac{125 x}{x - 5} \cdot \frac{4}{4} + \frac{C}{x - 5}$

Step 7.3.11

Write each expression with a common denominator of $4 (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.11.1

Multiply $\frac{125 x}{x - 5}$ by $\frac{4}{4}$ .

$y = \frac{x^{2} (x^{2} - 20 x + 150)}{4 (x - 5)} - \frac{125 x \cdot 4}{(x - 5) \cdot 4} + \frac{C}{x - 5}$

Step 7.3.11.2

Reorder the factors of $(x - 5) \cdot 4$ .

$y = \frac{x^{2} (x^{2} - 20 x + 150)}{4 (x - 5)} - \frac{125 x \cdot 4}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.12

Combine the numerators over the common denominator.

$y = \frac{x^{2} (x^{2} - 20 x + 150) - 125 x \cdot 4}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13

Simplify the numerator.

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Step 7.3.13.1

Factor $x$ out of $x^{2} (x^{2} - 20 x + 150) - 125 x \cdot 4$ .

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Step 7.3.13.1.1

Factor $x$ out of $x^{2} (x^{2} - 20 x + 150)$ .

$y = \frac{x (x (x^{2} - 20 x + 150)) - 125 x \cdot 4}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13.1.2

Factor $x$ out of $- 125 x \cdot 4$ .

$y = \frac{x (x (x^{2} - 20 x + 150)) + x (- 125 \cdot 4)}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13.1.3

Factor $x$ out of $x (x (x^{2} - 20 x + 150)) + x (- 125 \cdot 4)$ .

$y = \frac{x (x (x^{2} - 20 x + 150) - 125 \cdot 4)}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13.2

Apply the distributive property.

$y = \frac{x (x \cdot x^{2} + x (- 20 x) + x \cdot 150 - 125 \cdot 4)}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13.3

Simplify.

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Step 7.3.13.3.1

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

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Step 7.3.13.3.1.1

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

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Step 7.3.13.3.1.1.1

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

$y = \frac{x (x^{1} x^{2} + x (- 20 x) + x \cdot 150 - 125 \cdot 4)}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13.3.1.1.2

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

$y = \frac{x (x^{1 + 2} + x (- 20 x) + x \cdot 150 - 125 \cdot 4)}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13.3.1.2

Add $1$ and $2$ .

$y = \frac{x (x^{3} + x (- 20 x) + x \cdot 150 - 125 \cdot 4)}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13.3.2

Rewrite using the commutative property of multiplication.

$y = \frac{x (x^{3} - 20 x \cdot x + x \cdot 150 - 125 \cdot 4)}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13.3.3

Move $150$ to the left of $x$ .

$y = \frac{x (x^{3} - 20 x \cdot x + 150 \cdot x - 125 \cdot 4)}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13.4

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

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Step 7.3.13.4.1

Move $x$ .

$y = \frac{x (x^{3} - 20 (x \cdot x) + 150 \cdot x - 125 \cdot 4)}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13.4.2

Multiply $x$ by $x$ .

$y = \frac{x (x^{3} - 20 x^{2} + 150 \cdot x - 125 \cdot 4)}{4 (x - 5)} + \frac{C}{x - 5}$

$y = \frac{x (x^{3} - 20 x^{2} + 150 x - 125 \cdot 4)}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13.5

Multiply $- 125$ by $4$ .

$y = \frac{x (x^{3} - 20 x^{2} + 150 x - 500)}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.13.6

Factor $x^{3} - 20 x^{2} + 150 x - 500$ using the rational roots test.

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Step 7.3.13.6.1

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.

$p = \pm 1, \pm 500, \pm 2, \pm 250, \pm 4, \pm 125, \pm 5, \pm 100, \pm 10, \pm 50, \pm 20, \pm 25$

$q = \pm 1$

Step 7.3.13.6.2

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

$\pm 1, \pm 500, \pm 2, \pm 250, \pm 4, \pm 125, \pm 5, \pm 100, \pm 10, \pm 50, \pm 20, \pm 25$

Step 7.3.13.6.3

Substitute $10$ and simplify the expression. In this case, the expression is equal to $0$ so $10$ is a root of the polynomial.

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Step 7.3.13.6.3.1

Substitute $10$ into the polynomial.

$10^{3} - 20 \cdot 10^{2} + 150 \cdot 10 - 500$

Step 7.3.13.6.3.2

Raise $10$ to the power of $3$ .

$1000 - 20 \cdot 10^{2} + 150 \cdot 10 - 500$

Step 7.3.13.6.3.3

Raise $10$ to the power of $2$ .

$1000 - 20 \cdot 100 + 150 \cdot 10 - 500$

Step 7.3.13.6.3.4

Multiply $- 20$ by $100$ .

$1000 - 2000 + 150 \cdot 10 - 500$

Step 7.3.13.6.3.5

Subtract $2000$ from $1000$ .

$- 1000 + 150 \cdot 10 - 500$

Step 7.3.13.6.3.6

Multiply $150$ by $10$ .

$- 1000 + 1500 - 500$

Step 7.3.13.6.3.7

Add $- 1000$ and $1500$ .

$500 - 500$

Step 7.3.13.6.3.8

Subtract $500$ from $500$ .

$0$

Step 7.3.13.6.4

Since $10$ is a known root, divide the polynomial by $x - 10$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 20 x^{2} + 150 x - 500}{x - 10}$

Step 7.3.13.6.5

Divide $x^{3} - 20 x^{2} + 150 x - 500$ by $x - 10$ .

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Step 7.3.13.6.5.1

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

$x$

$10$

$x^{3}$

$20 x^{2}$

$150 x$

$500$

Step 7.3.13.6.5.2

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

					$x^{2}$
	$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$

Step 7.3.13.6.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			+	$x^{3}$	-	$10 x^{2}$

Step 7.3.13.6.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 10 x^{2}$

				$x^{2}$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			-	$x^{3}$	+	$10 x^{2}$

Step 7.3.13.6.5.5

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

				$x^{2}$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			-	$x^{3}$	+	$10 x^{2}$
					-	$10 x^{2}$

Step 7.3.13.6.5.6

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

				$x^{2}$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			-	$x^{3}$	+	$10 x^{2}$
					-	$10 x^{2}$	+	$150 x$

Step 7.3.13.6.5.7

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			-	$x^{3}$	+	$10 x^{2}$
					-	$10 x^{2}$	+	$150 x$

Step 7.3.13.6.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			-	$x^{3}$	+	$10 x^{2}$
					-	$10 x^{2}$	+	$150 x$
					-	$10 x^{2}$	+	$100 x$

Step 7.3.13.6.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} + 100 x$

				$x^{2}$	-	$10 x$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			-	$x^{3}$	+	$10 x^{2}$
					-	$10 x^{2}$	+	$150 x$
					+	$10 x^{2}$	-	$100 x$

Step 7.3.13.6.5.10

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

				$x^{2}$	-	$10 x$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			-	$x^{3}$	+	$10 x^{2}$
					-	$10 x^{2}$	+	$150 x$
					+	$10 x^{2}$	-	$100 x$
							+	$50 x$

Step 7.3.13.6.5.11

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

				$x^{2}$	-	$10 x$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			-	$x^{3}$	+	$10 x^{2}$
					-	$10 x^{2}$	+	$150 x$
					+	$10 x^{2}$	-	$100 x$
							+	$50 x$	-	$500$

Step 7.3.13.6.5.12

Divide the highest order term in the dividend $50 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$	+	$50$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			-	$x^{3}$	+	$10 x^{2}$
					-	$10 x^{2}$	+	$150 x$
					+	$10 x^{2}$	-	$100 x$
							+	$50 x$	-	$500$

Step 7.3.13.6.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$	+	$50$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			-	$x^{3}$	+	$10 x^{2}$
					-	$10 x^{2}$	+	$150 x$
					+	$10 x^{2}$	-	$100 x$
							+	$50 x$	-	$500$
							+	$50 x$	-	$500$

Step 7.3.13.6.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $50 x - 500$

				$x^{2}$	-	$10 x$	+	$50$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			-	$x^{3}$	+	$10 x^{2}$
					-	$10 x^{2}$	+	$150 x$
					+	$10 x^{2}$	-	$100 x$
							+	$50 x$	-	$500$
							-	$50 x$	+	$500$

Step 7.3.13.6.5.15

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

				$x^{2}$	-	$10 x$	+	$50$
$x$	-	$10$		$x^{3}$	-	$20 x^{2}$	+	$150 x$	-	$500$
			-	$x^{3}$	+	$10 x^{2}$
					-	$10 x^{2}$	+	$150 x$
					+	$10 x^{2}$	-	$100 x$
							+	$50 x$	-	$500$
							-	$50 x$	+	$500$
										$0$

Step 7.3.13.6.5.16

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

$x^{2} - 10 x + 50$

Step 7.3.13.6.6

Write $x^{3} - 20 x^{2} + 150 x - 500$ as a set of factors.

$y = \frac{x ((x - 10) (x^{2} - 10 x + 50))}{4 (x - 5)} + \frac{C}{x - 5}$

$y = \frac{x (x - 10) (x^{2} - 10 x + 50)}{4 (x - 5)} + \frac{C}{x - 5}$

Step 7.3.14

To write $\frac{C}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{x (x - 10) (x^{2} - 10 x + 50)}{4 (x - 5)} + \frac{C}{x - 5} \cdot \frac{4}{4}$

Step 7.3.15

Write each expression with a common denominator of $4 (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.15.1

Multiply $\frac{C}{x - 5}$ by $\frac{4}{4}$ .

$y = \frac{x (x - 10) (x^{2} - 10 x + 50)}{4 (x - 5)} + \frac{C \cdot 4}{(x - 5) \cdot 4}$

Step 7.3.15.2

Reorder the factors of $(x - 5) \cdot 4$ .

$y = \frac{x (x - 10) (x^{2} - 10 x + 50)}{4 (x - 5)} + \frac{C \cdot 4}{4 (x - 5)}$

Step 7.3.16

Combine the numerators over the common denominator.

$y = \frac{x (x - 10) (x^{2} - 10 x + 50) + C \cdot 4}{4 (x - 5)}$

Step 7.3.17

Simplify the numerator.

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Step 7.3.17.1

Apply the distributive property.

$y = \frac{(x \cdot x + x \cdot - 10) (x^{2} - 10 x + 50) + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.2

Multiply $x$ by $x$ .

$y = \frac{(x^{2} + x \cdot - 10) (x^{2} - 10 x + 50) + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.3

Move $- 10$ to the left of $x$ .

$y = \frac{(x^{2} - 10 \cdot x) (x^{2} - 10 x + 50) + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.4

Expand $(x^{2} - 10 x) (x^{2} - 10 x + 50)$ by multiplying each term in the first expression by each term in the second expression.

$y = \frac{x^{2} x^{2} + x^{2} (- 10 x) + x^{2} \cdot 50 - 10 x \cdot x^{2} - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5

Simplify each term.

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Step 7.3.17.5.1

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

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Step 7.3.17.5.1.1

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

$y = \frac{x^{2 + 2} + x^{2} (- 10 x) + x^{2} \cdot 50 - 10 x \cdot x^{2} - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.1.2

Add $2$ and $2$ .

$y = \frac{x^{4} + x^{2} (- 10 x) + x^{2} \cdot 50 - 10 x \cdot x^{2} - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.2

Rewrite using the commutative property of multiplication.

$y = \frac{x^{4} - 10 x^{2} x + x^{2} \cdot 50 - 10 x \cdot x^{2} - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.3

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

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Step 7.3.17.5.3.1

Move $x$ .

$y = \frac{x^{4} - 10 (x \cdot x^{2}) + x^{2} \cdot 50 - 10 x \cdot x^{2} - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.3.2

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

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Step 7.3.17.5.3.2.1

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

$y = \frac{x^{4} - 10 (x^{1} x^{2}) + x^{2} \cdot 50 - 10 x \cdot x^{2} - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.3.2.2

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

$y = \frac{x^{4} - 10 x^{1 + 2} + x^{2} \cdot 50 - 10 x \cdot x^{2} - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.3.3

Add $1$ and $2$ .

$y = \frac{x^{4} - 10 x^{3} + x^{2} \cdot 50 - 10 x \cdot x^{2} - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.4

Move $50$ to the left of $x^{2}$ .

$y = \frac{x^{4} - 10 x^{3} + 50 \cdot x^{2} - 10 x \cdot x^{2} - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.5

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

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Step 7.3.17.5.5.1

Move $x^{2}$ .

$y = \frac{x^{4} - 10 x^{3} + 50 x^{2} - 10 (x^{2} x) - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.5.2

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

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Step 7.3.17.5.5.2.1

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

$y = \frac{x^{4} - 10 x^{3} + 50 x^{2} - 10 (x^{2} x^{1}) - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.5.2.2

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

$y = \frac{x^{4} - 10 x^{3} + 50 x^{2} - 10 x^{2 + 1} - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.5.3

Add $2$ and $1$ .

$y = \frac{x^{4} - 10 x^{3} + 50 x^{2} - 10 x^{3} - 10 x (- 10 x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.6

Rewrite using the commutative property of multiplication.

$y = \frac{x^{4} - 10 x^{3} + 50 x^{2} - 10 x^{3} - 10 \cdot - 10 x \cdot x - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.7

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

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Step 7.3.17.5.7.1

Move $x$ .

$y = \frac{x^{4} - 10 x^{3} + 50 x^{2} - 10 x^{3} - 10 \cdot - 10 (x \cdot x) - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.7.2

Multiply $x$ by $x$ .

$y = \frac{x^{4} - 10 x^{3} + 50 x^{2} - 10 x^{3} - 10 \cdot - 10 x^{2} - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.8

Multiply $- 10$ by $- 10$ .

$y = \frac{x^{4} - 10 x^{3} + 50 x^{2} - 10 x^{3} + 100 x^{2} - 10 x \cdot 50 + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.5.9

Multiply $50$ by $- 10$ .

$y = \frac{x^{4} - 10 x^{3} + 50 x^{2} - 10 x^{3} + 100 x^{2} - 500 x + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.6

Subtract $10 x^{3}$ from $- 10 x^{3}$ .

$y = \frac{x^{4} - 20 x^{3} + 50 x^{2} + 100 x^{2} - 500 x + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.7

Add $50 x^{2}$ and $100 x^{2}$ .

$y = \frac{x^{4} - 20 x^{3} + 150 x^{2} - 500 x + C \cdot 4}{4 (x - 5)}$

Step 7.3.17.8

Move $4$ to the left of $C$ .

$y = \frac{x^{4} - 20 x^{3} + 150 x^{2} - 500 x + 4 C}{4 (x - 5)}$