Finite Math Examples

$0.1 x^{3} - 0.4 x^{2} - 1.1 x + 3$

Step 1

Write $0.1 x^{3} - 0.4 x^{2} - 1.1 x + 3$ as an equation.

$y = 0.1 x^{3} - 0.4 x^{2} - 1.1 x + 3$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = 0.1 x^{3} - 0.4 x^{2} - 1.1 x + 3$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $0.1 x^{3} - 0.4 x^{2} - 1.1 x + 3 = 0$ .

$0.1 x^{3} - 0.4 x^{2} - 1.1 x + 3 = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $0.1$ out of $0.1 x^{3} - 0.4 x^{2} - 1.1 x + 3$ .

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Step 2.2.2.1.1

Factor $0.1$ out of $0.1 x^{3}$ .

$0.1 (x^{3}) - 0.4 x^{2} - 1.1 x + 3 = 0$

Step 2.2.2.1.2

Factor $0.1$ out of $- 0.4 x^{2}$ .

$0.1 (x^{3}) + 0.1 (- 4 x^{2}) - 1.1 x + 3 = 0$

Step 2.2.2.1.3

Factor $0.1$ out of $- 1.1 x$ .

$0.1 (x^{3}) + 0.1 (- 4 x^{2}) + 0.1 (- 11 x) + 3 = 0$

Step 2.2.2.1.4

Factor $0.1$ out of $3$ .

$0.1 x^{3} + 0.1 (- 4 x^{2}) + 0.1 (- 11 x) + 0.1 \cdot 30 = 0$

Step 2.2.2.1.5

Factor $0.1$ out of $0.1 x^{3} + 0.1 (- 4 x^{2})$ .

$0.1 (x^{3} - 4 x^{2}) + 0.1 (- 11 x) + 0.1 \cdot 30 = 0$

Step 2.2.2.1.6

Factor $0.1$ out of $0.1 (x^{3} - 4 x^{2}) + 0.1 (- 11 x)$ .

$0.1 (x^{3} - 4 x^{2} - 11 x) + 0.1 \cdot 30 = 0$

Step 2.2.2.1.7

Factor $0.1$ out of $0.1 (x^{3} - 4 x^{2} - 11 x) + 0.1 \cdot 30$ .

$0.1 (x^{3} - 4 x^{2} - 11 x + 30) = 0$

Step 2.2.2.2

Factor $x^{3} - 4 x^{2} - 11 x + 30$ using the rational roots test.

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Step 2.2.2.2.1

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.

$p = \pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6$

$q = \pm 1$

Step 2.2.2.2.2

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

$\pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6$

Step 2.2.2.2.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 2.2.2.2.3.1

Substitute $2$ into the polynomial.

$2^{3} - 4 \cdot 2^{2} - 11 \cdot 2 + 30$

Step 2.2.2.2.3.2

Raise $2$ to the power of $3$ .

$8 - 4 \cdot 2^{2} - 11 \cdot 2 + 30$

Step 2.2.2.2.3.3

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

$8 - 4 \cdot 4 - 11 \cdot 2 + 30$

Step 2.2.2.2.3.4

Multiply $- 4$ by $4$ .

$8 - 16 - 11 \cdot 2 + 30$

Step 2.2.2.2.3.5

Subtract $16$ from $8$ .

$- 8 - 11 \cdot 2 + 30$

Step 2.2.2.2.3.6

Multiply $- 11$ by $2$ .

$- 8 - 22 + 30$

Step 2.2.2.2.3.7

Subtract $22$ from $- 8$ .

$- 30 + 30$

Step 2.2.2.2.3.8

Add $- 30$ and $30$ .

$0$

Step 2.2.2.2.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 4 x^{2} - 11 x + 30}{x - 2}$

Step 2.2.2.2.5

Divide $x^{3} - 4 x^{2} - 11 x + 30$ by $x - 2$ .

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Step 2.2.2.2.5.1

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

$x$

$2$

$x^{3}$

$4 x^{2}$

$11 x$

$30$

Step 2.2.2.2.5.2

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

					$x^{2}$
	$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$

Step 2.2.2.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			+	$x^{3}$	-	$2 x^{2}$

Step 2.2.2.2.5.4

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$

Step 2.2.2.2.5.5

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$

Step 2.2.2.2.5.6

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$11 x$

Step 2.2.2.2.5.7

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

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$11 x$

Step 2.2.2.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$11 x$
					-	$2 x^{2}$	+	$4 x$

Step 2.2.2.2.5.9

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

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$4 x$

Step 2.2.2.2.5.10

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

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$4 x$
							-	$15 x$

Step 2.2.2.2.5.11

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

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$4 x$
							-	$15 x$	+	$30$

Step 2.2.2.2.5.12

Divide the highest order term in the dividend $- 15 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$2 x$	-	$15$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$4 x$
							-	$15 x$	+	$30$

Step 2.2.2.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$	-	$15$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$4 x$
							-	$15 x$	+	$30$
							-	$15 x$	+	$30$

Step 2.2.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x + 30$

				$x^{2}$	-	$2 x$	-	$15$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$4 x$
							-	$15 x$	+	$30$
							+	$15 x$	-	$30$

Step 2.2.2.2.5.15

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

				$x^{2}$	-	$2 x$	-	$15$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$11 x$
					+	$2 x^{2}$	-	$4 x$
							-	$15 x$	+	$30$
							+	$15 x$	-	$30$
										$0$

Step 2.2.2.2.5.16

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

$x^{2} - 2 x - 15$

Step 2.2.2.2.6

Write $x^{3} - 4 x^{2} - 11 x + 30$ as a set of factors.

$0.1 ((x - 2) (x^{2} - 2 x - 15)) = 0$

Step 2.2.2.3

Factor.

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Step 2.2.2.3.1

Factor $x^{2} - 2 x - 15$ using the AC method.

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Step 2.2.2.3.1.1

Factor $x^{2} - 2 x - 15$ using the AC method.

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Step 2.2.2.3.1.1.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 2.2.2.3.1.1.2

Write the factored form using these integers.

$0.1 ((x - 2) ((x - 5) (x + 3))) = 0$

Step 2.2.2.3.1.2

Remove unnecessary parentheses.

$0.1 ((x - 2) (x - 5) (x + 3)) = 0$

Step 2.2.2.3.2

Remove unnecessary parentheses.

$0.1 (x - 2) (x - 5) (x + 3) = 0$

Step 2.2.3

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

$x - 2 = 0$

$x - 5 = 0$

$x + 3 = 0$

Step 2.2.4

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

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Step 2.2.4.1

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

$x - 2 = 0$

Step 2.2.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.2.5

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

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Step 2.2.5.1

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

$x - 5 = 0$

Step 2.2.5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.2.6

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

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Step 2.2.6.1

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

$x + 3 = 0$

Step 2.2.6.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.2.7

The final solution is all the values that make $0.1 (x - 2) (x - 5) (x + 3) = 0$ true.

$x = 2, 5, - 3$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(2, 0), (5, 0), (- 3, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = 0.1 {(0)}^{3} - 0.4 {(0)}^{2} - 1.1 \cdot 0 + 3$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = 0.1 \cdot 0^{3} - 0.4 {(0)}^{2} - 1.1 \cdot 0 + 3$

Step 3.2.2

Remove parentheses.

$y = 0.1 \cdot 0^{3} - 0.4 \cdot 0^{2} - 1.1 \cdot 0 + 3$

Step 3.2.3

Remove parentheses.

$y = 0.1 {(0)}^{3} - 0.4 {(0)}^{2} - 1.1 \cdot 0 + 3$

Step 3.2.4

Simplify $0.1 {(0)}^{3} - 0.4 {(0)}^{2} - 1.1 \cdot 0 + 3$ .

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

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

$y = 0.1 \cdot 0 - 0.4 {(0)}^{2} - 1.1 \cdot 0 + 3$

Step 3.2.4.1.2

Multiply $0.1$ by $0$ .

$y = 0 - 0.4 {(0)}^{2} - 1.1 \cdot 0 + 3$

Step 3.2.4.1.3

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

$y = 0 - 0.4 \cdot 0 - 1.1 \cdot 0 + 3$

Step 3.2.4.1.4

Multiply $- 0.4$ by $0$ .

$y = 0 + 0 - 1.1 \cdot 0 + 3$

Step 3.2.4.1.5

Multiply $- 1.1$ by $0$ .

$y = 0 + 0 + 0 + 3$

Step 3.2.4.2

Simplify by adding numbers.

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Step 3.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 3$

Step 3.2.4.2.2

Add $0$ and $0$ .

$y = 0 + 3$

Step 3.2.4.2.3

Add $0$ and $3$ .

$y = 3$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, 3)$

Step 4

List the intersections.

x-intercept(s): $(2, 0), (5, 0), (- 3, 0)$

y-intercept(s): $(0, 3)$

Step 5