Calculus Examples

$f (x) = \frac{15 x^{3} + 35 x^{2} - 100 x}{56 x - 2 x^{2} - 4 x^{3}}$

Step 1

Set the denominator in $\frac{15 x^{3} + 35 x^{2} - 100 x}{56 x - 2 x^{2} - 4 x^{3}}$ equal to $0$ to find where the expression is undefined.

$56 x - 2 x^{2} - 4 x^{3} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$56 u - 2 u^{2} - 4 u^{3} = 0$

Step 2.1.2

Factor $2 u$ out of $56 u - 2 u^{2} - 4 u^{3}$ .

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Step 2.1.2.1

Factor $2 u$ out of $56 u$ .

$2 u (28) - 2 u^{2} - 4 u^{3} = 0$

Step 2.1.2.2

Factor $2 u$ out of $- 2 u^{2}$ .

$2 u (28) + 2 u (- u) - 4 u^{3} = 0$

Step 2.1.2.3

Factor $2 u$ out of $- 4 u^{3}$ .

$2 u (28) + 2 u (- u) + 2 u (- 2 u^{2}) = 0$

Step 2.1.2.4

Factor $2 u$ out of $2 u (28) + 2 u (- u)$ .

$2 u (28 - u) + 2 u (- 2 u^{2}) = 0$

Step 2.1.2.5

Factor $2 u$ out of $2 u (28 - u) + 2 u (- 2 u^{2})$ .

$2 u (28 - u - 2 u^{2}) = 0$

Step 2.1.3

Factor.

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Step 2.1.3.1

Factor by grouping.

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Step 2.1.3.1.1

Reorder terms.

$2 u (- 2 u^{2} - u + 28) = 0$

Step 2.1.3.1.2

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 $a \cdot c = - 2 \cdot 28 = - 56$ and whose sum is $b = - 1$ .

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Step 2.1.3.1.2.1

Factor $- 1$ out of $- u$ .

$2 u (- 2 u^{2} - (u) + 28) = 0$

Step 2.1.3.1.2.2

Rewrite $- 1$ as $7$ plus $- 8$

$2 u (- 2 u^{2} + (7 - 8) u + 28) = 0$

Step 2.1.3.1.2.3

Apply the distributive property.

$2 u (- 2 u^{2} + 7 u - 8 u + 28) = 0$

Step 2.1.3.1.3

Factor out the greatest common factor from each group.

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Step 2.1.3.1.3.1

Group the first two terms and the last two terms.

$2 u ((- 2 u^{2} + 7 u) - 8 u + 28) = 0$

Step 2.1.3.1.3.2

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

$2 u (u (- 2 u + 7) + 4 (- 2 u + 7)) = 0$

Step 2.1.3.1.4

Factor the polynomial by factoring out the greatest common factor, $- 2 u + 7$ .

$2 u ((- 2 u + 7) (u + 4)) = 0$

Step 2.1.3.2

Remove unnecessary parentheses.

$2 u (- 2 u + 7) (u + 4) = 0$

Step 2.1.4

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

$2 x (- 2 x + 7) (x + 4) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$- 2 x + 7 = 0$

$x + 4 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $- 2 x + 7$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $- 2 x + 7$ equal to $0$ .

$- 2 x + 7 = 0$

Step 2.4.2

Solve $- 2 x + 7 = 0$ for $x$ .

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Step 2.4.2.1

Subtract $7$ from both sides of the equation.

$- 2 x = - 7$

Step 2.4.2.2

Divide each term in $- 2 x = - 7$ by $- 2$ and simplify.

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Step 2.4.2.2.1

Divide each term in $- 2 x = - 7$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 7}{- 2}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 7}{- 2}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{- 2}$

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{7}{2}$

Step 2.5

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.6

The final solution is all the values that make $2 x (- 2 x + 7) (x + 4) = 0$ true.

$x = 0, \frac{7}{2}, - 4$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 4) \cup (- 4, 0) \cup (0, \frac{7}{2}) \cup (\frac{7}{2}, \infty)$

Set-Builder Notation:

${x | x \neq 0, \frac{7}{2}, - 4}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, - \frac{15}{4}) \cup (- \frac{15}{4}, - \frac{17}{6}) \cup (- \frac{17}{6}, - \frac{25}{14}) \cup (- \frac{25}{14}, \infty)$

Set-Builder Notation:

${y | y \neq - \frac{25}{14}, - \frac{17}{6}, - \frac{15}{4}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - 4) \cup (- 4, 0) \cup (0, \frac{7}{2}) \cup (\frac{7}{2}, \infty), {x | x \neq 0, \frac{7}{2}, - 4}$

Range: $(- \infty, - \frac{15}{4}) \cup (- \frac{15}{4}, - \frac{17}{6}) \cup (- \frac{17}{6}, - \frac{25}{14}) \cup (- \frac{25}{14}, \infty), {y | y \neq - \frac{25}{14}, - \frac{17}{6}, - \frac{15}{4}}$

Step 6