Finite Math Examples

$f (x) = x^{3} + 5 x^{2} - 35 x - 39$

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 $x$ .

$0 = x^{3} + 5 x^{2} - 35 x - 39$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{3} + 5 x^{2} - 35 x - 39 = 0$ .

$x^{3} + 5 x^{2} - 35 x - 39 = 0$

Step 1.2.2

Factor $x^{3} + 5 x^{2} - 35 x - 39$ 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.

$p = \pm 1, \pm 39, \pm 3, \pm 13$

$q = \pm 1$

Step 1.2.2.2

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

$\pm 1, \pm 39, \pm 3, \pm 13$

Step 1.2.2.3

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 1.2.2.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} + 5 {(- 1)}^{2} - 35 \cdot - 1 - 39$

Step 1.2.2.3.2

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

$- 1 + 5 {(- 1)}^{2} - 35 \cdot - 1 - 39$

Step 1.2.2.3.3

Raise $- 1$ to the power of $2$ .

$- 1 + 5 \cdot 1 - 35 \cdot - 1 - 39$

Step 1.2.2.3.4

Multiply $5$ by $1$ .

$- 1 + 5 - 35 \cdot - 1 - 39$

Step 1.2.2.3.5

Add $- 1$ and $5$ .

$4 - 35 \cdot - 1 - 39$

Step 1.2.2.3.6

Multiply $- 35$ by $- 1$ .

$4 + 35 - 39$

Step 1.2.2.3.7

Add $4$ and $35$ .

$39 - 39$

Step 1.2.2.3.8

Subtract $39$ from $39$ .

$0$

Step 1.2.2.4

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.

$\frac{x^{3} + 5 x^{2} - 35 x - 39}{x + 1}$

Step 1.2.2.5

Divide $x^{3} + 5 x^{2} - 35 x - 39$ by $x + 1$ .

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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 $0$ .

$x$

$1$

$x^{3}$

$5 x^{2}$

$35 x$

$39$

Step 1.2.2.5.2

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$

Step 1.2.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			+	$x^{3}$	+	$x^{2}$

Step 1.2.2.5.4

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			-	$x^{3}$	-	$x^{2}$

Step 1.2.2.5.5

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$

Step 1.2.2.5.6

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$35 x$

Step 1.2.2.5.7

Divide the highest order term in the dividend $4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$4 x$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$35 x$

Step 1.2.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$4 x$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$35 x$
					+	$4 x^{2}$	+	$4 x$

Step 1.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{2} + 4 x$

				$x^{2}$	+	$4 x$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$35 x$
					-	$4 x^{2}$	-	$4 x$

Step 1.2.2.5.10

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

				$x^{2}$	+	$4 x$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$35 x$
					-	$4 x^{2}$	-	$4 x$
							-	$39 x$

Step 1.2.2.5.11

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

				$x^{2}$	+	$4 x$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$35 x$
					-	$4 x^{2}$	-	$4 x$
							-	$39 x$	-	$39$

Step 1.2.2.5.12

Divide the highest order term in the dividend $- 39 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$4 x$	-	$39$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$35 x$
					-	$4 x^{2}$	-	$4 x$
							-	$39 x$	-	$39$

Step 1.2.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$4 x$	-	$39$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$35 x$
					-	$4 x^{2}$	-	$4 x$
							-	$39 x$	-	$39$
							-	$39 x$	-	$39$

Step 1.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 39 x - 39$

				$x^{2}$	+	$4 x$	-	$39$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$35 x$
					-	$4 x^{2}$	-	$4 x$
							-	$39 x$	-	$39$
							+	$39 x$	+	$39$

Step 1.2.2.5.15

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

				$x^{2}$	+	$4 x$	-	$39$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$35 x$	-	$39$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$35 x$
					-	$4 x^{2}$	-	$4 x$
							-	$39 x$	-	$39$
							+	$39 x$	+	$39$
										$0$

Step 1.2.2.5.16

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

$x^{2} + 4 x - 39$

Step 1.2.2.6

Write $x^{3} + 5 x^{2} - 35 x - 39$ as a set of factors.

$(x + 1) (x^{2} + 4 x - 39) = 0$

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 $0$ .

$x + 1 = 0$

$x^{2} + 4 x - 39 = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x + 1 = 0$

Step 1.2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.5

Set $x^{2} + 4 x - 39$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x^{2} + 4 x - 39$ equal to $0$ .

$x^{2} + 4 x - 39 = 0$

Step 1.2.5.2

Solve $x^{2} + 4 x - 39 = 0$ 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 $a = 1$ , $b = 4$ , and $c = - 39$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 39)}}{2 \cdot 1}$

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

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 39}}{2 \cdot 1}$

Step 1.2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 39$ .

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Step 1.2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 39}}{2 \cdot 1}$

Step 1.2.5.2.3.1.2.2

Multiply $- 4$ by $- 39$ .

$x = \frac{- 4 \pm \sqrt{16 + 156}}{2 \cdot 1}$

Step 1.2.5.2.3.1.3

Add $16$ and $156$ .

$x = \frac{- 4 \pm \sqrt{172}}{2 \cdot 1}$

Step 1.2.5.2.3.1.4

Rewrite $172$ as $2^{2} \cdot 43$ .

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Step 1.2.5.2.3.1.4.1

Factor $4$ out of $172$ .

$x = \frac{- 4 \pm \sqrt{4 (43)}}{2 \cdot 1}$

Step 1.2.5.2.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 43}}{2 \cdot 1}$

Step 1.2.5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{43}}{2 \cdot 1}$

Step 1.2.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{43}}{2}$

Step 1.2.5.2.3.3

Simplify $\frac{- 4 \pm 2 \sqrt{43}}{2}$ .

$x = - 2 \pm \sqrt{43}$

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

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 39}}{2 \cdot 1}$

Step 1.2.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 39$ .

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Step 1.2.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 39}}{2 \cdot 1}$

Step 1.2.5.2.4.1.2.2

Multiply $- 4$ by $- 39$ .

$x = \frac{- 4 \pm \sqrt{16 + 156}}{2 \cdot 1}$

Step 1.2.5.2.4.1.3

Add $16$ and $156$ .

$x = \frac{- 4 \pm \sqrt{172}}{2 \cdot 1}$

Step 1.2.5.2.4.1.4

Rewrite $172$ as $2^{2} \cdot 43$ .

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Step 1.2.5.2.4.1.4.1

Factor $4$ out of $172$ .

$x = \frac{- 4 \pm \sqrt{4 (43)}}{2 \cdot 1}$

Step 1.2.5.2.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 43}}{2 \cdot 1}$

Step 1.2.5.2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{43}}{2 \cdot 1}$

Step 1.2.5.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{43}}{2}$

Step 1.2.5.2.4.3

Simplify $\frac{- 4 \pm 2 \sqrt{43}}{2}$ .

$x = - 2 \pm \sqrt{43}$

Step 1.2.5.2.4.4

Change the $\pm$ to $+$ .

$x = - 2 + \sqrt{43}$

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

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 39}}{2 \cdot 1}$

Step 1.2.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 39$ .

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Step 1.2.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 39}}{2 \cdot 1}$

Step 1.2.5.2.5.1.2.2

Multiply $- 4$ by $- 39$ .

$x = \frac{- 4 \pm \sqrt{16 + 156}}{2 \cdot 1}$

Step 1.2.5.2.5.1.3

Add $16$ and $156$ .

$x = \frac{- 4 \pm \sqrt{172}}{2 \cdot 1}$

Step 1.2.5.2.5.1.4

Rewrite $172$ as $2^{2} \cdot 43$ .

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Step 1.2.5.2.5.1.4.1

Factor $4$ out of $172$ .

$x = \frac{- 4 \pm \sqrt{4 (43)}}{2 \cdot 1}$

Step 1.2.5.2.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 43}}{2 \cdot 1}$

Step 1.2.5.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{43}}{2 \cdot 1}$

Step 1.2.5.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{43}}{2}$

Step 1.2.5.2.5.3

Simplify $\frac{- 4 \pm 2 \sqrt{43}}{2}$ .

$x = - 2 \pm \sqrt{43}$

Step 1.2.5.2.5.4

Change the $\pm$ to $-$ .

$x = - 2 - \sqrt{43}$

Step 1.2.5.2.6

The final answer is the combination of both solutions.

$x = - 2 + \sqrt{43}, - 2 - \sqrt{43}$

Step 1.2.6

The final solution is all the values that make $(x + 1) (x^{2} + 4 x - 39) = 0$ true.

$x = - 1, - 2 + \sqrt{43}, - 2 - \sqrt{43}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 1, 0), (- 2 + \sqrt{43}, 0), (- 2 - \sqrt{43}, 0)$

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 $y$ .

$y = {(0)}^{3} + 5 {(0)}^{2} - 35 \cdot 0 - 39$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{3} + 5 {(0)}^{2} - 35 \cdot 0 - 39$

Step 2.2.2

Remove parentheses.

$y = 0^{3} + 5 \cdot 0^{2} - 35 \cdot 0 - 39$

Step 2.2.3

Remove parentheses.

$y = {(0)}^{3} + 5 {(0)}^{2} - 35 \cdot 0 - 39$

Step 2.2.4

Simplify ${(0)}^{3} + 5 {(0)}^{2} - 35 \cdot 0 - 39$ .

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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 $0$ .

$y = 0 + 5 {(0)}^{2} - 35 \cdot 0 - 39$

Step 2.2.4.1.2

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

$y = 0 + 5 \cdot 0 - 35 \cdot 0 - 39$

Step 2.2.4.1.3

Multiply $5$ by $0$ .

$y = 0 + 0 - 35 \cdot 0 - 39$

Step 2.2.4.1.4

Multiply $- 35$ by $0$ .

$y = 0 + 0 + 0 - 39$

Step 2.2.4.2

Simplify by adding and subtracting.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 - 39$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0 - 39$

Step 2.2.4.2.3

Subtract $39$ from $0$ .

$y = - 39$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 39)$

Step 3

List the intersections.

x-intercept(s): $(- 1, 0), (- 2 + \sqrt{43}, 0), (- 2 - \sqrt{43}, 0)$

y-intercept(s): $(0, - 39)$

Step 4