Precalculus Examples

$f (x) = 3 x^{3} + 4 x^{2} + 4 x + 1$

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 $x$ .

$0 = 3 x^{3} + 4 x^{2} + 4 x + 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 x^{3} + 4 x^{2} + 4 x + 1 = 0$ .

$3 x^{3} + 4 x^{2} + 4 x + 1 = 0$

Step 1.2.2

Factor $3 x^{3} + 4 x^{2} + 4 x + 1$ using the rational roots test.

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Step 1.2.2.1

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.

$p = \pm 1$

$q = \pm 1, \pm 3$

Step 1.2.2.2

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

$\pm 1, \pm 0.\overline{3}$

Step 1.2.2.3

Substitute $- 0.\overline{3}$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.\overline{3}$ is a root of the polynomial.

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Step 1.2.2.3.1

Substitute $- 0.\overline{3}$ into the polynomial.

$3 {(- 0.\overline{3})}^{3} + 4 {(- 0.\overline{3})}^{2} + 4 \cdot - 0.\overline{3} + 1$

Step 1.2.2.3.2

Raise $- 0.\overline{3}$ to the power of $3$ .

$3 \cdot - 0.\overline{037} + 4 {(- 0.\overline{3})}^{2} + 4 \cdot - 0.\overline{3} + 1$

Step 1.2.2.3.3

Multiply $3$ by $- 0.\overline{037}$ .

$- 0.\overline{1} + 4 {(- 0.\overline{3})}^{2} + 4 \cdot - 0.\overline{3} + 1$

Step 1.2.2.3.4

Raise $- 0.\overline{3}$ to the power of $2$ .

$- 0.\overline{1} + 4 \cdot 0.\overline{1} + 4 \cdot - 0.\overline{3} + 1$

Step 1.2.2.3.5

Multiply $4$ by $0.\overline{1}$ .

$- 0.\overline{1} + 0.\overline{4} + 4 \cdot - 0.\overline{3} + 1$

Step 1.2.2.3.6

Add $- 0.\overline{1}$ and $0.\overline{4}$ .

$0.\overline{3} + 4 \cdot - 0.\overline{3} + 1$

Step 1.2.2.3.7

Multiply $4$ by $- 0.\overline{3}$ .

$0.\overline{3} - 1.\overline{3} + 1$

Step 1.2.2.3.8

Subtract $1.\overline{3}$ from $0.\overline{3}$ .

$- 1 + 1$

Step 1.2.2.3.9

Add $- 1$ and $1$ .

$0$

Step 1.2.2.4

Since $- 0.\overline{3}$ is a known root, divide the polynomial by $3 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{3} + 4 x^{2} + 4 x + 1}{3 x + 1}$

Step 1.2.2.5

Divide $3 x^{3} + 4 x^{2} + 4 x + 1$ by $3 x + 1$ .

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Step 1.2.2.5.1

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

$3 x$

$1$

$3 x^{3}$

$4 x^{2}$

$4 x$

$1$

Step 1.2.2.5.2

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

					$x^{2}$
	$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$

Step 1.2.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			+	$3 x^{3}$	+	$x^{2}$

Step 1.2.2.5.4

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

				$x^{2}$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			-	$3 x^{3}$	-	$x^{2}$

Step 1.2.2.5.5

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

				$x^{2}$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			-	$3 x^{3}$	-	$x^{2}$
					+	$3 x^{2}$

Step 1.2.2.5.6

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

				$x^{2}$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			-	$3 x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	+	$4 x$

Step 1.2.2.5.7

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $3 x$ .

				$x^{2}$	+	$x$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			-	$3 x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	+	$4 x$

Step 1.2.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			-	$3 x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	+	$4 x$
					+	$3 x^{2}$	+	$x$

Step 1.2.2.5.9

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

				$x^{2}$	+	$x$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			-	$3 x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	-	$x$

Step 1.2.2.5.10

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

				$x^{2}$	+	$x$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			-	$3 x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	-	$x$
							+	$3 x$

Step 1.2.2.5.11

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

				$x^{2}$	+	$x$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			-	$3 x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	-	$x$
							+	$3 x$	+	$1$

Step 1.2.2.5.12

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

				$x^{2}$	+	$x$	+	$1$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			-	$3 x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	-	$x$
							+	$3 x$	+	$1$

Step 1.2.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$	+	$1$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			-	$3 x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	-	$x$
							+	$3 x$	+	$1$
							+	$3 x$	+	$1$

Step 1.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $3 x + 1$

				$x^{2}$	+	$x$	+	$1$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			-	$3 x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	-	$x$
							+	$3 x$	+	$1$
							-	$3 x$	-	$1$

Step 1.2.2.5.15

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

				$x^{2}$	+	$x$	+	$1$
$3 x$	+	$1$		$3 x^{3}$	+	$4 x^{2}$	+	$4 x$	+	$1$
			-	$3 x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	-	$x$
							+	$3 x$	+	$1$
							-	$3 x$	-	$1$
										$0$

Step 1.2.2.5.16

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

$x^{2} + x + 1$

Step 1.2.2.6

Write $3 x^{3} + 4 x^{2} + 4 x + 1$ as a set of factors.

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

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 $0$ .

$3 x + 1 = 0$

$x^{2} + x + 1 = 0$

Step 1.2.4

Set $3 x + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Step 1.2.4.2

Solve $3 x + 1 = 0$ for $x$ .

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Step 1.2.4.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 1.2.4.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 1.2.4.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

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 $3$ .

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Step 1.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 1.2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 1.2.4.2.2.3

Simplify the right side.

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Step 1.2.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 1.2.5

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

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Step 1.2.5.1

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

$x^{2} + x + 1 = 0$

Step 1.2.5.2

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

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Step 1.2.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.5.2.2

Substitute the values $a = 1$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 1.2.5.2.3

Simplify.

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Step 1.2.5.2.3.1

Simplify the numerator.

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Step 1.2.5.2.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 1.2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 1.2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 1.2.5.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 1.2.5.2.3.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 1.2.5.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 1.2.5.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 1.2.5.2.3.1.6

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

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 1.2.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 1.2.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.5.2.4.1

Simplify the numerator.

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Step 1.2.5.2.4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 1.2.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 1.2.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 1.2.5.2.4.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 1.2.5.2.4.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 1.2.5.2.4.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 1.2.5.2.4.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 1.2.5.2.4.1.6

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

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 1.2.5.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 1.2.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{3}}{2}$

Step 1.2.5.2.4.4

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

$x = \frac{- 1 \cdot 1 + i \sqrt{3}}{2}$

Step 1.2.5.2.4.5

Factor $- 1$ out of $i \sqrt{3}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{3})}{2}$

Step 1.2.5.2.4.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{3})$ .

$x = \frac{- 1 (1 - i \sqrt{3})}{2}$

Step 1.2.5.2.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{3}}{2}$

Step 1.2.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.5.2.5.1

Simplify the numerator.

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Step 1.2.5.2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 1.2.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 1.2.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 1.2.5.2.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 1.2.5.2.5.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 1.2.5.2.5.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 1.2.5.2.5.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 1.2.5.2.5.1.6

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

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 1.2.5.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 1.2.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{3}}{2}$

Step 1.2.5.2.5.4

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

$x = \frac{- 1 \cdot 1 - i \sqrt{3}}{2}$

Step 1.2.5.2.5.5

Factor $- 1$ out of $- i \sqrt{3}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{3})}{2}$

Step 1.2.5.2.5.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{3})$ .

$x = \frac{- 1 (1 + i \sqrt{3})}{2}$

Step 1.2.5.2.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{3}}{2}$

Step 1.2.5.2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 1.2.6

The final solution is all the values that make $(3 x + 1) (x^{2} + x + 1) = 0$ true.

$x = - \frac{1}{3}, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{1}{3}, 0)$

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 $y$ .

$y = 3 {(0)}^{3} + 4 {(0)}^{2} + 4 (0) + 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 3 \cdot 0^{3} + 4 {(0)}^{2} + 4 (0) + 1$

Step 2.2.2

Remove parentheses.

$y = 3 \cdot 0^{3} + 4 \cdot 0^{2} + 4 (0) + 1$

Step 2.2.3

Remove parentheses.

$y = 3 {(0)}^{3} + 4 {(0)}^{2} + 4 (0) + 1$

Step 2.2.4

Simplify $3 {(0)}^{3} + 4 {(0)}^{2} + 4 (0) + 1$ .

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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 $0$ .

$y = 3 \cdot 0 + 4 {(0)}^{2} + 4 (0) + 1$

Step 2.2.4.1.2

Multiply $3$ by $0$ .

$y = 0 + 4 {(0)}^{2} + 4 (0) + 1$

Step 2.2.4.1.3

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

$y = 0 + 4 \cdot 0 + 4 (0) + 1$

Step 2.2.4.1.4

Multiply $4$ by $0$ .

$y = 0 + 0 + 4 (0) + 1$

Step 2.2.4.1.5

Multiply $4$ by $0$ .

$y = 0 + 0 + 0 + 1$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 1$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0 + 1$

Step 2.2.4.2.3

Add $0$ and $1$ .

$y = 1$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(- \frac{1}{3}, 0)$

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

Step 4