Precalculus Examples

$f (x) = 5 x^{3} - x^{2} + 5 x - 1$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 5 x^{3} - x^{2} + 5 x - 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $5 x^{3} - x^{2} + 5 x - 1 = 0$ .

$5 x^{3} - x^{2} + 5 x - 1 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor out the greatest common factor from each group.

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Step 1.2.2.1.1

Group the first two terms and the last two terms.

$(5 x^{3} - x^{2}) + 5 x - 1 = 0$

Step 1.2.2.1.2

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

$x^{2} (5 x - 1) + 1 (5 x - 1) = 0$

Step 1.2.2.2

Factor the polynomial by factoring out the greatest common factor, $5 x - 1$ .

$(5 x - 1) (x^{2} + 1) = 0$

Step 1.2.3

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

$5 x - 1 = 0$

$x^{2} + 1 = 0$

Step 1.2.4

Set $5 x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $5 x - 1$ equal to $0$ .

$5 x - 1 = 0$

Step 1.2.4.2

Solve $5 x - 1 = 0$ for $x$ .

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Step 1.2.4.2.1

Add $1$ to both sides of the equation.

$5 x = 1$

Step 1.2.4.2.2

Divide each term in $5 x = 1$ by $5$ and simplify.

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Step 1.2.4.2.2.1

Divide each term in $5 x = 1$ by $5$ .

$\frac{5 x}{5} = \frac{1}{5}$

Step 1.2.4.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{1}{5}$

Step 1.2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{5}$

Step 1.2.5

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 1.2.5.2

Solve $x^{2} + 1 = 0$ for $x$ .

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Step 1.2.5.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 1.2.5.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Step 1.2.5.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 1.2.5.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.5.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 1.2.5.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 1.2.5.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 1.2.6

The final solution is all the values that make $(5 x - 1) (x^{2} + 1) = 0$ true.

$x = \frac{1}{5}, i, - i$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{1}{5}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 5 {(0)}^{3} - {(0)}^{2} + 5 (0) - 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 5 \cdot 0^{3} - {(0)}^{2} + 5 (0) - 1$

Step 2.2.2

Remove parentheses.

$y = 5 \cdot 0^{3} - 0^{2} + 5 (0) - 1$

Step 2.2.3

Remove parentheses.

$y = 5 {(0)}^{3} - {(0)}^{2} + 5 (0) - 1$

Step 2.2.4

Simplify $5 {(0)}^{3} - {(0)}^{2} + 5 (0) - 1$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = 5 \cdot 0 - {(0)}^{2} + 5 (0) - 1$

Step 2.2.4.1.2

Multiply $5$ by $0$ .

$y = 0 - {(0)}^{2} + 5 (0) - 1$

Step 2.2.4.1.3

Raising $0$ to any positive power yields $0$ .

$y = 0 - 0 + 5 (0) - 1$

Step 2.2.4.1.4

Multiply $- 1$ by $0$ .

$y = 0 + 0 + 5 (0) - 1$

Step 2.2.4.1.5

Multiply $5$ by $0$ .

$y = 0 + 0 + 0 - 1$

Step 2.2.4.2

Simplify by adding and subtracting.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 - 1$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0 - 1$

Step 2.2.4.2.3

Subtract $1$ from $0$ .

$y = - 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1)$

Step 3

List the intersections.

x-intercept(s): $(\frac{1}{5}, 0)$

y-intercept(s): $(0, - 1)$

Step 4