Precalculus Examples

$f (x) = x^{3} - 5 x^{2} - 9 x + 45$

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 $x$ .

$0 = x^{3} - 5 x^{2} - 9 x + 45$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{3} - 5 x^{2} - 9 x + 45 = 0$ .

$x^{3} - 5 x^{2} - 9 x + 45 = 0$

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.

$(x^{3} - 5 x^{2}) - 9 x + 45 = 0$

Step 1.2.2.1.2

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

$x^{2} (x - 5) - 9 (x - 5) = 0$

Step 1.2.2.2

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

$(x - 5) (x^{2} - 9) = 0$

Step 1.2.2.3

Rewrite $9$ as $3^{2}$ .

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

Step 1.2.2.4

Factor.

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Step 1.2.2.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$(x - 5) ((x + 3) (x - 3)) = 0$

Step 1.2.2.4.2

Remove unnecessary parentheses.

$(x - 5) (x + 3) (x - 3) = 0$

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 $0$ .

$x - 5 = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x - 5 = 0$

Step 1.2.4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 1.2.5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 1.2.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.6

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 1.2.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.7

The final solution is all the values that make $(x - 5) (x + 3) (x - 3) = 0$ true.

$x = 5, - 3, 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(5, 0), (- 3, 0), (3, 0)$

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 $y$ .

$y = {(0)}^{3} - 5 {(0)}^{2} - 9 \cdot 0 + 45$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{3} - 5 {(0)}^{2} - 9 \cdot 0 + 45$

Step 2.2.2

Remove parentheses.

$y = 0^{3} - 5 \cdot 0^{2} - 9 \cdot 0 + 45$

Step 2.2.3

Remove parentheses.

$y = {(0)}^{3} - 5 {(0)}^{2} - 9 \cdot 0 + 45$

Step 2.2.4

Simplify ${(0)}^{3} - 5 {(0)}^{2} - 9 \cdot 0 + 45$ .

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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 $0$ .

$y = 0 - 5 {(0)}^{2} - 9 \cdot 0 + 45$

Step 2.2.4.1.2

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

$y = 0 - 5 \cdot 0 - 9 \cdot 0 + 45$

Step 2.2.4.1.3

Multiply $- 5$ by $0$ .

$y = 0 + 0 - 9 \cdot 0 + 45$

Step 2.2.4.1.4

Multiply $- 9$ by $0$ .

$y = 0 + 0 + 0 + 45$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 45$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0 + 45$

Step 2.2.4.2.3

Add $0$ and $45$ .

$y = 45$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 45)$

Step 3

List the intersections.

x-intercept(s): $(5, 0), (- 3, 0), (3, 0)$

y-intercept(s): $(0, 45)$

Step 4