Algebra Examples

$p (x) = x^{4} - d x^{3} + 8 x^{2} - 14 x + 16$

Step 1

Rewrite the equation as $x^{4} - d x^{3} + 8 x^{2} - 14 x + 16 = p x$ .

$x^{4} - d x^{3} + 8 x^{2} - 14 x + 16 = p x$

Step 2

Move all terms not containing $d$ to the right side of the equation.

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Step 2.1

Subtract $x^{4}$ from both sides of the equation.

$- d x^{3} + 8 x^{2} - 14 x + 16 = p x - x^{4}$

Step 2.2

Subtract $8 x^{2}$ from both sides of the equation.

$- d x^{3} - 14 x + 16 = p x - x^{4} - 8 x^{2}$

Step 2.3

Add $14 x$ to both sides of the equation.

$- d x^{3} + 16 = p x - x^{4} - 8 x^{2} + 14 x$

Step 2.4

Subtract $16$ from both sides of the equation.

$- d x^{3} = p x - x^{4} - 8 x^{2} + 14 x - 16$

Step 3

Divide each term in $- d x^{3} = p x - x^{4} - 8 x^{2} + 14 x - 16$ by $- x^{3}$ and simplify.

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Step 3.1

Divide each term in $- d x^{3} = p x - x^{4} - 8 x^{2} + 14 x - 16$ by $- x^{3}$ .

$\frac{- d x^{3}}{- x^{3}} = \frac{p x}{- x^{3}} + \frac{- x^{4}}{- x^{3}} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{d x^{3}}{x^{3}} = \frac{p x}{- x^{3}} + \frac{- x^{4}}{- x^{3}} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.2.2

Cancel the common factor of $x^{3}$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{d x^{3}}{x^{3}} = \frac{p x}{- x^{3}} + \frac{- x^{4}}{- x^{3}} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.2.2.2

Divide $d$ by $1$ .

$d = \frac{p x}{- x^{3}} + \frac{- x^{4}}{- x^{3}} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $x$ and $x^{3}$ .

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Step 3.3.1.1.1

Factor $x$ out of $p x$ .

$d = \frac{x p}{- x^{3}} + \frac{- x^{4}}{- x^{3}} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.1.2

Cancel the common factors.

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Step 3.3.1.1.2.1

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

$d = \frac{x p}{x (- x^{2})} + \frac{- x^{4}}{- x^{3}} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.1.2.2

Cancel the common factor.

$d = \frac{x p}{x (- x^{2})} + \frac{- x^{4}}{- x^{3}} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.1.2.3

Rewrite the expression.

$d = \frac{p}{- x^{2}} + \frac{- x^{4}}{- x^{3}} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.2

Move the negative in front of the fraction.

$d = - \frac{p}{x^{2}} + \frac{- x^{4}}{- x^{3}} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.3

Dividing two negative values results in a positive value.

$d = - \frac{p}{x^{2}} + \frac{x^{4}}{x^{3}} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.4

Cancel the common factor of $x^{4}$ and $x^{3}$ .

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Step 3.3.1.4.1

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

$d = - \frac{p}{x^{2}} + \frac{x^{3} x}{x^{3}} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.4.2

Cancel the common factors.

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Step 3.3.1.4.2.1

Multiply by $1$ .

$d = - \frac{p}{x^{2}} + \frac{x^{3} x}{x^{3} \cdot 1} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.4.2.2

Cancel the common factor.

$d = - \frac{p}{x^{2}} + \frac{x^{3} x}{x^{3} \cdot 1} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.4.2.3

Rewrite the expression.

$d = - \frac{p}{x^{2}} + \frac{x}{1} + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.4.2.4

Divide $x$ by $1$ .

$d = - \frac{p}{x^{2}} + x + \frac{- 8 x^{2}}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.5

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 3.3.1.5.1

Factor $x^{2}$ out of $- 8 x^{2}$ .

$d = - \frac{p}{x^{2}} + x + \frac{x^{2} \cdot - 8}{- x^{3}} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.5.2

Cancel the common factors.

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Step 3.3.1.5.2.1

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

$d = - \frac{p}{x^{2}} + x + \frac{x^{2} \cdot - 8}{x^{2} (- x)} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.5.2.2

Cancel the common factor.

$d = - \frac{p}{x^{2}} + x + \frac{x^{2} \cdot - 8}{x^{2} (- x)} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.5.2.3

Rewrite the expression.

$d = - \frac{p}{x^{2}} + x + \frac{- 8}{- x} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.6

Dividing two negative values results in a positive value.

$d = - \frac{p}{x^{2}} + x + \frac{8}{x} + \frac{14 x}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.7

Cancel the common factor of $x$ and $x^{3}$ .

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Step 3.3.1.7.1

Factor $x$ out of $14 x$ .

$d = - \frac{p}{x^{2}} + x + \frac{8}{x} + \frac{x \cdot 14}{- x^{3}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.7.2

Cancel the common factors.

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Step 3.3.1.7.2.1

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

$d = - \frac{p}{x^{2}} + x + \frac{8}{x} + \frac{x \cdot 14}{x (- x^{2})} + \frac{- 16}{- x^{3}}$

Step 3.3.1.7.2.2

Cancel the common factor.

$d = - \frac{p}{x^{2}} + x + \frac{8}{x} + \frac{x \cdot 14}{x (- x^{2})} + \frac{- 16}{- x^{3}}$

Step 3.3.1.7.2.3

Rewrite the expression.

$d = - \frac{p}{x^{2}} + x + \frac{8}{x} + \frac{14}{- x^{2}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.8

Move the negative in front of the fraction.

$d = - \frac{p}{x^{2}} + x + \frac{8}{x} - \frac{14}{x^{2}} + \frac{- 16}{- x^{3}}$

Step 3.3.1.9

Dividing two negative values results in a positive value.

$d = - \frac{p}{x^{2}} + x + \frac{8}{x} - \frac{14}{x^{2}} + \frac{16}{x^{3}}$