Finite Math Examples

$f (x) = x^{4} + 2 x^{3} - 10 x^{2} - 26 x + 33$

Step 1

Set $x^{4} + 2 x^{3} - 10 x^{2} - 26 x + 33$ equal to $0$ .

$x^{4} + 2 x^{3} - 10 x^{2} - 26 x + 33 = 0$

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 $x^{4} + 2 x^{3} - 10 x^{2} - 26 x + 33$ 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.

$p = \pm 1, \pm 33, \pm 3, \pm 11$

$q = \pm 1$

Step 2.1.1.2

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

$\pm 1, \pm 33, \pm 3, \pm 11$

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.

$1^{4} + 2 \cdot 1^{3} - 10 \cdot 1^{2} - 26 \cdot 1 + 33$

Step 2.1.1.3.2

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

$1 + 2 \cdot 1^{3} - 10 \cdot 1^{2} - 26 \cdot 1 + 33$

Step 2.1.1.3.3

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

$1 + 2 \cdot 1 - 10 \cdot 1^{2} - 26 \cdot 1 + 33$

Step 2.1.1.3.4

Multiply $2$ by $1$ .

$1 + 2 - 10 \cdot 1^{2} - 26 \cdot 1 + 33$

Step 2.1.1.3.5

Add $1$ and $2$ .

$3 - 10 \cdot 1^{2} - 26 \cdot 1 + 33$

Step 2.1.1.3.6

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

$3 - 10 \cdot 1 - 26 \cdot 1 + 33$

Step 2.1.1.3.7

Multiply $- 10$ by $1$ .

$3 - 10 - 26 \cdot 1 + 33$

Step 2.1.1.3.8

Subtract $10$ from $3$ .

$- 7 - 26 \cdot 1 + 33$

Step 2.1.1.3.9

Multiply $- 26$ by $1$ .

$- 7 - 26 + 33$

Step 2.1.1.3.10

Subtract $26$ from $- 7$ .

$- 33 + 33$

Step 2.1.1.3.11

Add $- 33$ and $33$ .

$0$

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.

$\frac{x^{4} + 2 x^{3} - 10 x^{2} - 26 x + 33}{x - 1}$

Step 2.1.1.5

Divide $x^{4} + 2 x^{3} - 10 x^{2} - 26 x + 33$ 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 $0$ .

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

Step 2.1.1.5.2

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

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

Step 2.1.1.5.4

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

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

Step 2.1.1.5.5

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

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

Step 2.1.1.5.6

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

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

Step 2.1.1.5.7

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

Step 2.1.1.5.9

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

Step 2.1.1.5.10

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

$7 x^{2}$

Step 2.1.1.5.11

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

$7 x^{2}$

$26 x$

Step 2.1.1.5.12

Divide the highest order term in the dividend $- 7 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

$7 x^{2}$

$26 x$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

$7 x^{2}$

$26 x$

$7 x^{2}$

$7 x$

Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{2} + 7 x$

$x^{3}$

$3 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

$7 x^{2}$

$26 x$

$7 x^{2}$

$7 x$

Step 2.1.1.5.15

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

$x^{3}$

$3 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

$7 x^{2}$

$26 x$

$7 x^{2}$

$7 x$

$33 x$

Step 2.1.1.5.16

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

$x^{3}$

$3 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

$7 x^{2}$

$26 x$

$7 x^{2}$

$7 x$

$33 x$

$33$

Step 2.1.1.5.17

Divide the highest order term in the dividend $- 33 x$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$7 x$

$33$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

$7 x^{2}$

$26 x$

$7 x^{2}$

$7 x$

$33 x$

$33$

Step 2.1.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$7 x$

$33$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

$7 x^{2}$

$26 x$

$7 x^{2}$

$7 x$

$33 x$

$33$

$33 x$

$33$

Step 2.1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 33 x + 33$

$x^{3}$

$3 x^{2}$

$7 x$

$33$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

$7 x^{2}$

$26 x$

$7 x^{2}$

$7 x$

$33 x$

$33$

$33 x$

$33$

Step 2.1.1.5.20

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

$x^{3}$

$3 x^{2}$

$7 x$

$33$

$x$

$1$

$x^{4}$

$2 x^{3}$

$10 x^{2}$

$26 x$

$33$

$x^{4}$

$x^{3}$

$3 x^{3}$

$10 x^{2}$

$3 x^{3}$

$3 x^{2}$

$7 x^{2}$

$26 x$

$7 x^{2}$

$7 x$

$33 x$

$33$

$33 x$

$33$

$0$

Step 2.1.1.5.21

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

$x^{3} + 3 x^{2} - 7 x - 33$

Step 2.1.1.6

Write $x^{4} + 2 x^{3} - 10 x^{2} - 26 x + 33$ as a set of factors.

$(x - 1) (x^{3} + 3 x^{2} - 7 x - 33) = 0$

Step 2.1.2

Factor $x^{3} + 3 x^{2} - 7 x - 33$ using the rational roots test.

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Step 2.1.2.1

Factor $x^{3} + 3 x^{2} - 7 x - 33$ 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.

$p = \pm 1, \pm 33, \pm 3, \pm 11$

$q = \pm 1$

Step 2.1.2.1.2

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

$\pm 1, \pm 33, \pm 3, \pm 11$

Step 2.1.2.1.3

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

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Step 2.1.2.1.3.1

Substitute $3$ into the polynomial.

$3^{3} + 3 \cdot 3^{2} - 7 \cdot 3 - 33$

Step 2.1.2.1.3.2

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

$27 + 3 \cdot 3^{2} - 7 \cdot 3 - 33$

Step 2.1.2.1.3.3

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

$27 + 3 \cdot 9 - 7 \cdot 3 - 33$

Step 2.1.2.1.3.4

Multiply $3$ by $9$ .

$27 + 27 - 7 \cdot 3 - 33$

Step 2.1.2.1.3.5

Add $27$ and $27$ .

$54 - 7 \cdot 3 - 33$

Step 2.1.2.1.3.6

Multiply $- 7$ by $3$ .

$54 - 21 - 33$

Step 2.1.2.1.3.7

Subtract $21$ from $54$ .

$33 - 33$

Step 2.1.2.1.3.8

Subtract $33$ from $33$ .

$0$

Step 2.1.2.1.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 3 x^{2} - 7 x - 33}{x - 3}$

Step 2.1.2.1.5

Divide $x^{3} + 3 x^{2} - 7 x - 33$ by $x - 3$ .

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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 $0$ .

$x$

$3$

$x^{3}$

$3 x^{2}$

$7 x$

$33$

Step 2.1.2.1.5.2

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$

Step 2.1.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			+	$x^{3}$	-	$3 x^{2}$

Step 2.1.2.1.5.4

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			-	$x^{3}$	+	$3 x^{2}$

Step 2.1.2.1.5.5

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$

Step 2.1.2.1.5.6

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$7 x$

Step 2.1.2.1.5.7

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$6 x$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$7 x$

Step 2.1.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					+	$6 x^{2}$	-	$18 x$

Step 2.1.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} - 18 x$

				$x^{2}$	+	$6 x$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$18 x$

Step 2.1.2.1.5.10

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

				$x^{2}$	+	$6 x$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$18 x$
							+	$11 x$

Step 2.1.2.1.5.11

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

				$x^{2}$	+	$6 x$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$18 x$
							+	$11 x$	-	$33$

Step 2.1.2.1.5.12

Divide the highest order term in the dividend $11 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$6 x$	+	$11$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$18 x$
							+	$11 x$	-	$33$

Step 2.1.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$	+	$11$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$18 x$
							+	$11 x$	-	$33$
							+	$11 x$	-	$33$

Step 2.1.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $11 x - 33$

				$x^{2}$	+	$6 x$	+	$11$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$18 x$
							+	$11 x$	-	$33$
							-	$11 x$	+	$33$

Step 2.1.2.1.5.15

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

				$x^{2}$	+	$6 x$	+	$11$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$7 x$	-	$33$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$18 x$
							+	$11 x$	-	$33$
							-	$11 x$	+	$33$
										$0$

Step 2.1.2.1.5.16

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

$x^{2} + 6 x + 11$

Step 2.1.2.1.6

Write $x^{3} + 3 x^{2} - 7 x - 33$ as a set of factors.

$(x - 1) ((x - 3) (x^{2} + 6 x + 11)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$(x - 1) (x - 3) (x^{2} + 6 x + 11) = 0$

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 $0$ .

$x - 1 = 0$

$x - 3 = 0$

$x^{2} + 6 x + 11 = 0$

Step 2.3

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

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Step 2.3.1

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

$x - 1 = 0$

Step 2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.4

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

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Step 2.4.1

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

$x - 3 = 0$

Step 2.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.5

Set $x^{2} + 6 x + 11$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} + 6 x + 11$ equal to $0$ .

$x^{2} + 6 x + 11 = 0$

Step 2.5.2

Solve $x^{2} + 6 x + 11 = 0$ for $x$ .

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 2.5.2.2

Substitute the values $a = 1$ , $b = 6$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 11)}}{2 \cdot 1}$

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 11}}{2 \cdot 1}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 11$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 11}}{2 \cdot 1}$

Step 2.5.2.3.1.2.2

Multiply $- 4$ by $11$ .

$x = \frac{- 6 \pm \sqrt{36 - 44}}{2 \cdot 1}$

Step 2.5.2.3.1.3

Subtract $44$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 2.5.2.3.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 2.5.2.3.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.5.2.3.1.6

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

$x = \frac{- 6 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.5.2.3.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.5.2.3.1.7.1

Factor $4$ out of $8$ .

$x = \frac{- 6 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 2.5.2.3.1.7.2

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

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

Step 2.5.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 6 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 2.5.2.3.1.9

Move $2$ to the left of $i$ .

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

Step 2.5.2.3.2

Multiply $2$ by $1$ .

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

Step 2.5.2.3.3

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

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

Step 2.5.2.4

The final answer is the combination of both solutions.

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

Step 2.6

The final solution is all the values that make $(x - 1) (x - 3) (x^{2} + 6 x + 11) = 0$ true.

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

Step 3