Finite Math Examples

$x^{4} + 7 x^{2} + 18 x + 10$

Step 1

Set $x^{4} + 7 x^{2} + 18 x + 10$ equal to $0$ .

$x^{4} + 7 x^{2} + 18 x + 10 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $x^{4} + 7 x^{2} + 18 x + 10$ using the rational roots test.

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Step 2.1.1.1

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.

$p = \pm 1, \pm 10, \pm 2, \pm 5$

$q = \pm 1$

Step 2.1.1.2

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

$\pm 1, \pm 10, \pm 2, \pm 5$

Step 2.1.1.3

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 2.1.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} + 7 {(- 1)}^{2} + 18 \cdot - 1 + 10$

Step 2.1.1.3.2

Raise $- 1$ to the power of $4$ .

$1 + 7 {(- 1)}^{2} + 18 \cdot - 1 + 10$

Step 2.1.1.3.3

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

$1 + 7 \cdot 1 + 18 \cdot - 1 + 10$

Step 2.1.1.3.4

Multiply $7$ by $1$ .

$1 + 7 + 18 \cdot - 1 + 10$

Step 2.1.1.3.5

Add $1$ and $7$ .

$8 + 18 \cdot - 1 + 10$

Step 2.1.1.3.6

Multiply $18$ by $- 1$ .

$8 - 18 + 10$

Step 2.1.1.3.7

Subtract $18$ from $8$ .

$- 10 + 10$

Step 2.1.1.3.8

Add $- 10$ and $10$ .

$0$

Step 2.1.1.4

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.

$\frac{x^{4} + 7 x^{2} + 18 x + 10}{x + 1}$

Step 2.1.1.5

Divide $x^{4} + 7 x^{2} + 18 x + 10$ by $x + 1$ .

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Step 2.1.1.5.1

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

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

Step 2.1.1.5.2

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

Step 2.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + x^{3}$

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

Step 2.1.1.5.5

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

Step 2.1.1.5.6

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

Step 2.1.1.5.7

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

Step 2.1.1.5.9

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

Step 2.1.1.5.10

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

$8 x^{2}$

Step 2.1.1.5.11

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

$8 x^{2}$

$18 x$

Step 2.1.1.5.12

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2}$

$8 x$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

$8 x^{2}$

$18 x$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$8 x$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

$8 x^{2}$

$18 x$

$8 x^{2}$

$8 x$

Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} + 8 x$

$x^{3}$

$x^{2}$

$8 x$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

$8 x^{2}$

$18 x$

$8 x^{2}$

$8 x$

Step 2.1.1.5.15

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

$x^{3}$

$x^{2}$

$8 x$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

$8 x^{2}$

$18 x$

$8 x^{2}$

$8 x$

$10 x$

Step 2.1.1.5.16

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

$x^{3}$

$x^{2}$

$8 x$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

$8 x^{2}$

$18 x$

$8 x^{2}$

$8 x$

$10 x$

$10$

Step 2.1.1.5.17

Divide the highest order term in the dividend $10 x$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2}$

$8 x$

$10$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

$8 x^{2}$

$18 x$

$8 x^{2}$

$8 x$

$10 x$

$10$

Step 2.1.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$8 x$

$10$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

$8 x^{2}$

$18 x$

$8 x^{2}$

$8 x$

$10 x$

$10$

$10 x$

$10$

Step 2.1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $10 x + 10$

$x^{3}$

$x^{2}$

$8 x$

$10$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

$8 x^{2}$

$18 x$

$8 x^{2}$

$8 x$

$10 x$

$10$

$10 x$

$10$

Step 2.1.1.5.20

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

$x^{3}$

$x^{2}$

$8 x$

$10$

$x$

$1$

$x^{4}$

$0 x^{3}$

$7 x^{2}$

$18 x$

$10$

$x^{4}$

$x^{3}$

$7 x^{2}$

$x^{3}$

$x^{2}$

$8 x^{2}$

$18 x$

$8 x^{2}$

$8 x$

$10 x$

$10$

$10 x$

$10$

$0$

Step 2.1.1.5.21

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

$x^{3} - x^{2} + 8 x + 10$

Step 2.1.1.6

Write $x^{4} + 7 x^{2} + 18 x + 10$ as a set of factors.

$(x + 1) (x^{3} - x^{2} + 8 x + 10) = 0$

Step 2.1.2

Factor $x^{3} - x^{2} + 8 x + 10$ using the rational roots test.

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Step 2.1.2.1

Factor $x^{3} - x^{2} + 8 x + 10$ using the rational roots test.

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Step 2.1.2.1.1

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.

$p = \pm 1, \pm 10, \pm 2, \pm 5$

$q = \pm 1$

Step 2.1.2.1.2

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

$\pm 1, \pm 10, \pm 2, \pm 5$

Step 2.1.2.1.3

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 2.1.2.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} - {(- 1)}^{2} + 8 \cdot - 1 + 10$

Step 2.1.2.1.3.2

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

$- 1 - {(- 1)}^{2} + 8 \cdot - 1 + 10$

Step 2.1.2.1.3.3

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

$- 1 - 1 \cdot 1 + 8 \cdot - 1 + 10$

Step 2.1.2.1.3.4

Multiply $- 1$ by $1$ .

$- 1 - 1 + 8 \cdot - 1 + 10$

Step 2.1.2.1.3.5

Subtract $1$ from $- 1$ .

$- 2 + 8 \cdot - 1 + 10$

Step 2.1.2.1.3.6

Multiply $8$ by $- 1$ .

$- 2 - 8 + 10$

Step 2.1.2.1.3.7

Subtract $8$ from $- 2$ .

$- 10 + 10$

Step 2.1.2.1.3.8

Add $- 10$ and $10$ .

$0$

Step 2.1.2.1.4

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.

$\frac{x^{3} - x^{2} + 8 x + 10}{x + 1}$

Step 2.1.2.1.5

Divide $x^{3} - x^{2} + 8 x + 10$ by $x + 1$ .

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Step 2.1.2.1.5.1

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

$x$

$1$

$x^{3}$

$x^{2}$

$8 x$

$10$

Step 2.1.2.1.5.2

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$

Step 2.1.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			+	$x^{3}$	+	$x^{2}$

Step 2.1.2.1.5.4

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			-	$x^{3}$	-	$x^{2}$

Step 2.1.2.1.5.5

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			-	$x^{3}$	-	$x^{2}$
					-	$2 x^{2}$

Step 2.1.2.1.5.6

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			-	$x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	+	$8 x$

Step 2.1.2.1.5.7

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

				$x^{2}$	-	$2 x$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			-	$x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	+	$8 x$

Step 2.1.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			-	$x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	+	$8 x$
					-	$2 x^{2}$	-	$2 x$

Step 2.1.2.1.5.9

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

				$x^{2}$	-	$2 x$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			-	$x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	+	$8 x$
					+	$2 x^{2}$	+	$2 x$

Step 2.1.2.1.5.10

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

				$x^{2}$	-	$2 x$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			-	$x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	+	$8 x$
					+	$2 x^{2}$	+	$2 x$
							+	$10 x$

Step 2.1.2.1.5.11

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

				$x^{2}$	-	$2 x$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			-	$x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	+	$8 x$
					+	$2 x^{2}$	+	$2 x$
							+	$10 x$	+	$10$

Step 2.1.2.1.5.12

Divide the highest order term in the dividend $10 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$2 x$	+	$10$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			-	$x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	+	$8 x$
					+	$2 x^{2}$	+	$2 x$
							+	$10 x$	+	$10$

Step 2.1.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$	+	$10$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			-	$x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	+	$8 x$
					+	$2 x^{2}$	+	$2 x$
							+	$10 x$	+	$10$
							+	$10 x$	+	$10$

Step 2.1.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $10 x + 10$

				$x^{2}$	-	$2 x$	+	$10$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			-	$x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	+	$8 x$
					+	$2 x^{2}$	+	$2 x$
							+	$10 x$	+	$10$
							-	$10 x$	-	$10$

Step 2.1.2.1.5.15

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

				$x^{2}$	-	$2 x$	+	$10$
$x$	+	$1$		$x^{3}$	-	$x^{2}$	+	$8 x$	+	$10$
			-	$x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	+	$8 x$
					+	$2 x^{2}$	+	$2 x$
							+	$10 x$	+	$10$
							-	$10 x$	-	$10$
										$0$

Step 2.1.2.1.5.16

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

$x^{2} - 2 x + 10$

Step 2.1.2.1.6

Write $x^{3} - x^{2} + 8 x + 10$ as a set of factors.

$(x + 1) ((x + 1) (x^{2} - 2 x + 10)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$(x + 1) (x + 1) (x^{2} - 2 x + 10) = 0$

Step 2.1.3

Combine like factors.

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Step 2.1.3.1

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

$(x + 1) (x + 1) (x^{2} - 2 x + 10) = 0$

Step 2.1.3.2

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

$(x + 1) (x + 1) (x^{2} - 2 x + 10) = 0$

Step 2.1.3.3

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

${(x + 1)}^{1 + 1} (x^{2} - 2 x + 10) = 0$

Step 2.1.3.4

Add $1$ and $1$ .

${(x + 1)}^{2} (x^{2} - 2 x + 10) = 0$

Step 2.2

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

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

$x^{2} - 2 x + 10 = 0$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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Step 2.3.2.1

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

$x + 1 = 0$

Step 2.3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.4

Set $x^{2} - 2 x + 10$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2} - 2 x + 10$ equal to $0$ .

$x^{2} - 2 x + 10 = 0$

Step 2.4.2

Solve $x^{2} - 2 x + 10 = 0$ for $x$ .

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Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

Substitute the values $a = 1$ , $b = - 2$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 10)}}{2 \cdot 1}$

Step 2.4.2.3

Simplify.

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Step 2.4.2.3.1

Simplify the numerator.

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Step 2.4.2.3.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 2.4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 2.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 10}}{2 \cdot 1}$

Step 2.4.2.3.1.2.2

Multiply $- 4$ by $10$ .

$x = \frac{2 \pm \sqrt{4 - 40}}{2 \cdot 1}$

Step 2.4.2.3.1.3

Subtract $40$ from $4$ .

$x = \frac{2 \pm \sqrt{- 36}}{2 \cdot 1}$

Step 2.4.2.3.1.4

Rewrite $- 36$ as $- 1 (36)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 36}}{2 \cdot 1}$

Step 2.4.2.3.1.5

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{36}}{2 \cdot 1}$

Step 2.4.2.3.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{36}}{2 \cdot 1}$

Step 2.4.2.3.1.7

Rewrite $36$ as $6^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{6^{2}}}{2 \cdot 1}$

Step 2.4.2.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 6}{2 \cdot 1}$

Step 2.4.2.3.1.9

Move $6$ to the left of $i$ .

$x = \frac{2 \pm 6 i}{2 \cdot 1}$

Step 2.4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 6 i}{2}$

Step 2.4.2.3.3

Simplify $\frac{2 \pm 6 i}{2}$ .

$x = 1 \pm 3 i$

Step 2.4.2.4

The final answer is the combination of both solutions.

$x = 1 + 3 i, 1 - 3 i$

Step 2.5

The final solution is all the values that make ${(x + 1)}^{2} (x^{2} - 2 x + 10) = 0$ true.

$x = - 1, 1 + 3 i, 1 - 3 i$

Step 3