Precalculus Examples

$p (x) = - \frac{1}{2} x^{3} + \frac{3}{2} x - 1$

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Combine $x^{3}$ and $\frac{1}{2}$ .

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

Step 1.2.2.2

Combine $\frac{3}{2}$ and $x$ .

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

Step 1.2.3

Multiply each term in $- \frac{x^{3}}{2} + \frac{3 x}{2} - 1 = 0$ by $2$ to eliminate the fractions.

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Step 1.2.3.1

Multiply each term in $- \frac{x^{3}}{2} + \frac{3 x}{2} - 1 = 0$ by $2$ .

$- \frac{x^{3}}{2} \cdot 2 + \frac{3 x}{2} \cdot 2 - 1 \cdot 2 = 0 \cdot 2$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify each term.

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Step 1.2.3.2.1.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1.1

Move the leading negative in $- \frac{x^{3}}{2}$ into the numerator.

$\frac{- x^{3}}{2} \cdot 2 + \frac{3 x}{2} \cdot 2 - 1 \cdot 2 = 0 \cdot 2$

Step 1.2.3.2.1.1.2

Cancel the common factor.

$\frac{- x^{3}}{2} \cdot 2 + \frac{3 x}{2} \cdot 2 - 1 \cdot 2 = 0 \cdot 2$

Step 1.2.3.2.1.1.3

Rewrite the expression.

$- x^{3} + \frac{3 x}{2} \cdot 2 - 1 \cdot 2 = 0 \cdot 2$

Step 1.2.3.2.1.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.2.1

Cancel the common factor.

$- x^{3} + \frac{3 x}{2} \cdot 2 - 1 \cdot 2 = 0 \cdot 2$

Step 1.2.3.2.1.2.2

Rewrite the expression.

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

Step 1.2.3.2.1.3

Multiply $- 1$ by $2$ .

$- x^{3} + 3 x - 2 = 0 \cdot 2$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Multiply $0$ by $2$ .

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

Step 1.2.4

Factor the left side of the equation.

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Step 1.2.4.1

Factor $- 1$ out of $- x^{3} + 3 x - 2$ .

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Step 1.2.4.1.1

Factor $- 1$ out of $- x^{3}$ .

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

Step 1.2.4.1.2

Factor $- 1$ out of $3 x$ .

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

Step 1.2.4.1.3

Rewrite $- 2$ as $- 1 (2)$ .

$- (x^{3}) - (- 3 x) - 1 \cdot 2 = 0$

Step 1.2.4.1.4

Factor $- 1$ out of $- (x^{3}) - (- 3 x)$ .

$- (x^{3} - 3 x) - 1 \cdot 2 = 0$

Step 1.2.4.1.5

Factor $- 1$ out of $- (x^{3} - 3 x) - 1 (2)$ .

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

Step 1.2.4.2

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

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Step 1.2.4.2.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1$

Step 1.2.4.2.2

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

$\pm 1, \pm 2$

Step 1.2.4.2.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 1.2.4.2.3.1

Substitute $1$ into the polynomial.

$1^{3} - 3 \cdot 1 + 2$

Step 1.2.4.2.3.2

Raise $1$ to the power of $3$ .

$1 - 3 \cdot 1 + 2$

Step 1.2.4.2.3.3

Multiply $- 3$ by $1$ .

$1 - 3 + 2$

Step 1.2.4.2.3.4

Subtract $3$ from $1$ .

$- 2 + 2$

Step 1.2.4.2.3.5

Add $- 2$ and $2$ .

$0$

Step 1.2.4.2.4

Since $1$ is a known root, divide the polynomial by $x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 3 x + 2}{x - 1}$

Step 1.2.4.2.5

Divide $x^{3} - 3 x + 2$ by $x - 1$ .

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Step 1.2.4.2.5.1

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

$x$

$1$

$x^{3}$

$0 x^{2}$

$3 x$

$2$

Step 1.2.4.2.5.2

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

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

Step 1.2.4.2.5.3

Multiply the new quotient term by the divisor.

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

Step 1.2.4.2.5.4

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

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

Step 1.2.4.2.5.5

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

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

Step 1.2.4.2.5.6

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

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

Step 1.2.4.2.5.7

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

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

Step 1.2.4.2.5.8

Multiply the new quotient term by the divisor.

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

Step 1.2.4.2.5.9

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

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

Step 1.2.4.2.5.10

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

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

Step 1.2.4.2.5.11

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

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

Step 1.2.4.2.5.12

Divide the highest order term in the dividend $- 2 x$ by the highest order term in divisor $x$ .

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

Step 1.2.4.2.5.13

Multiply the new quotient term by the divisor.

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

Step 1.2.4.2.5.14

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

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

Step 1.2.4.2.5.15

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

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

Step 1.2.4.2.5.16

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

$x^{2} + x - 2$

Step 1.2.4.2.6

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

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

Step 1.2.4.3

Factor $x^{2} + x - 2$ using the AC method.

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Step 1.2.4.3.1

Factor $x^{2} + x - 2$ using the AC method.

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Step 1.2.4.3.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 1.2.4.3.1.2

Write the factored form using these integers.

$- ((x - 1) ((x - 1) (x + 2))) = 0$

Step 1.2.4.3.2

Remove unnecessary parentheses.

$- ((x - 1) (x - 1) (x + 2)) = 0$

Step 1.2.4.4

Factor.

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Step 1.2.4.4.1

Combine like factors.

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Step 1.2.4.4.1.1

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

$- ((x - 1) (x - 1) (x + 2)) = 0$

Step 1.2.4.4.1.2

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

$- ((x - 1) (x - 1) (x + 2)) = 0$

Step 1.2.4.4.1.3

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

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

Step 1.2.4.4.1.4

Add $1$ and $1$ .

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

Step 1.2.4.4.2

Remove unnecessary parentheses.

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

Step 1.2.5

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

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

$x + 2 = 0$

Step 1.2.6

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set ${(x - 1)}^{2}$ equal to $0$ .

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

Step 1.2.6.2

Solve ${(x - 1)}^{2} = 0$ for $x$ .

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Step 1.2.6.2.1

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

$x - 1 = 0$

Step 1.2.6.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.7

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

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Step 1.2.7.1

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

$x + 2 = 0$

Step 1.2.7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.8

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

$x = 1, - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(1, 0), (- 2, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Multiply $\frac{3}{2}$ by $0$ .

$y = - \frac{1}{2} \cdot 0^{3} + \frac{3}{2} \cdot 0 - 1$

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

Simplify $- \frac{1}{2} \cdot {(0)}^{3} + \frac{3}{2} \cdot (0) - 1$ .

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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 $0$ .

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

Step 2.2.4.1.2

Multiply $- \frac{1}{2} \cdot 0$ .

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Step 2.2.4.1.2.1

Multiply $0$ by $- 1$ .

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

Step 2.2.4.1.2.2

Multiply $0$ by $\frac{1}{2}$ .

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

Step 2.2.4.1.3

Multiply $\frac{3}{2}$ by $0$ .

$y = 0 + 0 - 1$

Step 2.2.4.2

Simplify by adding and subtracting.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 - 1$

Step 2.2.4.2.2

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): $(1, 0), (- 2, 0)$

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

Step 4