Finite Math Examples

$3 x^{4} - 37 x^{3} + 111 x^{2} - 45 x + 4$

Step 1

Factor $3 x^{4} - 37 x^{3} + 111 x^{2} - 45 x + 4$ using the rational roots test.

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Step 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.

$p = \pm 1, \pm 4, \pm 2$

$q = \pm 1, \pm 3$

Step 1.2

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

$\pm 1, \pm 0.\overline{3}, \pm 4, \pm 1.\overline{3}, \pm 2, \pm 0.\overline{6}$

Step 1.3

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 1.3.1

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

$3 \cdot {0.\overline{3}}^{4} - 37 \cdot {0.\overline{3}}^{3} + 111 \cdot {0.\overline{3}}^{2} - 45 \cdot 0.\overline{3} + 4$

Step 1.3.2

Raise $0.\overline{3}$ to the power of $4$ .

$3 \cdot 0.01234567 - 37 \cdot {0.\overline{3}}^{3} + 111 \cdot {0.\overline{3}}^{2} - 45 \cdot 0.\overline{3} + 4$

Step 1.3.3

Multiply $3$ by $0.01234567$ .

$0.\overline{037} - 37 \cdot {0.\overline{3}}^{3} + 111 \cdot {0.\overline{3}}^{2} - 45 \cdot 0.\overline{3} + 4$

Step 1.3.4

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

$0.\overline{037} - 37 \cdot 0.\overline{037} + 111 \cdot {0.\overline{3}}^{2} - 45 \cdot 0.\overline{3} + 4$

Step 1.3.5

Multiply $- 37$ by $0.\overline{037}$ .

$0.\overline{037} - 1.37037037 + 111 \cdot {0.\overline{3}}^{2} - 45 \cdot 0.\overline{3} + 4$

Step 1.3.6

Subtract $1.37037037$ from $0.\overline{037}$ .

$- 1.33333333 + 111 \cdot {0.\overline{3}}^{2} - 45 \cdot 0.\overline{3} + 4$

Step 1.3.7

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

$- 1.33333333 + 111 \cdot 0.\overline{1} - 45 \cdot 0.\overline{3} + 4$

Step 1.3.8

Multiply $111$ by $0.\overline{1}$ .

$- 1.33333333 + 12.33333333 - 45 \cdot 0.\overline{3} + 4$

Step 1.3.9

Add $- 1.33333333$ and $12.33333333$ .

$11 - 45 \cdot 0.\overline{3} + 4$

Step 1.3.10

Multiply $- 45$ by $0.\overline{3}$ .

$11 - 15 + 4$

Step 1.3.11

Subtract $15$ from $11$ .

$- 4 + 4$

Step 1.3.12

Add $- 4$ and $4$ .

$0$

Step 1.4

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.

$\frac{3 x^{4} - 37 x^{3} + 111 x^{2} - 45 x + 4}{3 x - 1}$

Step 1.5

Divide $3 x^{4} - 37 x^{3} + 111 x^{2} - 45 x + 4$ by $3 x - 1$ .

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Step 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 $0$ .

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

Step 1.5.2

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $3 x$ .

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

Step 1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

Step 1.5.4

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

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

Step 1.5.5

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

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

Step 1.5.6

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

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

Step 1.5.7

Divide the highest order term in the dividend $- 36 x^{3}$ by the highest order term in divisor $3 x$ .

$x^{3}$

$12 x^{2}$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

Step 1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 36 x^{3} + 12 x^{2}$

$x^{3}$

$12 x^{2}$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

Step 1.5.10

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

$x^{3}$

$12 x^{2}$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

$99 x^{2}$

Step 1.5.11

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

$x^{3}$

$12 x^{2}$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

$99 x^{2}$

$45 x$

Step 1.5.12

Divide the highest order term in the dividend $99 x^{2}$ by the highest order term in divisor $3 x$ .

$x^{3}$

$12 x^{2}$

$33 x$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

$99 x^{2}$

$45 x$

Step 1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$33 x$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

$99 x^{2}$

$45 x$

$99 x^{2}$

$33 x$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $99 x^{2} - 33 x$

$x^{3}$

$12 x^{2}$

$33 x$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

$99 x^{2}$

$45 x$

$99 x^{2}$

$33 x$

Step 1.5.15

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

$x^{3}$

$12 x^{2}$

$33 x$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

$99 x^{2}$

$45 x$

$99 x^{2}$

$33 x$

$12 x$

Step 1.5.16

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

$x^{3}$

$12 x^{2}$

$33 x$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

$99 x^{2}$

$45 x$

$99 x^{2}$

$33 x$

$12 x$

$4$

Step 1.5.17

Divide the highest order term in the dividend $- 12 x$ by the highest order term in divisor $3 x$ .

$x^{3}$

$12 x^{2}$

$33 x$

$4$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

$99 x^{2}$

$45 x$

$99 x^{2}$

$33 x$

$12 x$

$4$

Step 1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$33 x$

$4$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

$99 x^{2}$

$45 x$

$99 x^{2}$

$33 x$

$12 x$

$4$

$12 x$

$4$

Step 1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x + 4$

$x^{3}$

$12 x^{2}$

$33 x$

$4$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

$99 x^{2}$

$45 x$

$99 x^{2}$

$33 x$

$12 x$

$4$

$12 x$

$4$

Step 1.5.20

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

$x^{3}$

$12 x^{2}$

$33 x$

$4$

$3 x$

$1$

$3 x^{4}$

$37 x^{3}$

$111 x^{2}$

$45 x$

$4$

$3 x^{4}$

$x^{3}$

$36 x^{3}$

$111 x^{2}$

$36 x^{3}$

$12 x^{2}$

$99 x^{2}$

$45 x$

$99 x^{2}$

$33 x$

$12 x$

$4$

$12 x$

$4$

$0$

Step 1.5.21

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

$x^{3} - 12 x^{2} + 33 x - 4$

Step 1.6

Write $3 x^{4} - 37 x^{3} + 111 x^{2} - 45 x + 4$ as a set of factors.

$(3 x - 1) (x^{3} - 12 x^{2} + 33 x - 4)$

Step 2

Factor $x^{3} - 12 x^{2} + 33 x - 4$ using the rational roots test.

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Step 2.1

Factor $x^{3} - 12 x^{2} + 33 x - 4$ using the rational roots test.

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Step 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.

$p = \pm 1, \pm 4, \pm 2$

$q = \pm 1$

Step 2.1.2

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

$\pm 1, \pm 4, \pm 2$

Step 2.1.3

Substitute $4$ and simplify the expression. In this case, the expression is equal to $0$ so $4$ is a root of the polynomial.

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Step 2.1.3.1

Substitute $4$ into the polynomial.

$4^{3} - 12 \cdot 4^{2} + 33 \cdot 4 - 4$

Step 2.1.3.2

Raise $4$ to the power of $3$ .

$64 - 12 \cdot 4^{2} + 33 \cdot 4 - 4$

Step 2.1.3.3

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

$64 - 12 \cdot 16 + 33 \cdot 4 - 4$

Step 2.1.3.4

Multiply $- 12$ by $16$ .

$64 - 192 + 33 \cdot 4 - 4$

Step 2.1.3.5

Subtract $192$ from $64$ .

$- 128 + 33 \cdot 4 - 4$

Step 2.1.3.6

Multiply $33$ by $4$ .

$- 128 + 132 - 4$

Step 2.1.3.7

Add $- 128$ and $132$ .

$4 - 4$

Step 2.1.3.8

Subtract $4$ from $4$ .

$0$

Step 2.1.4

Since $4$ is a known root, divide the polynomial by $x - 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 12 x^{2} + 33 x - 4}{x - 4}$

Step 2.1.5

Divide $x^{3} - 12 x^{2} + 33 x - 4$ by $x - 4$ .

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Step 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 $0$ .

$x$

$4$

$x^{3}$

$12 x^{2}$

$33 x$

$4$

Step 2.1.5.2

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

					$x^{2}$
	$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			+	$x^{3}$	-	$4 x^{2}$

Step 2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 4 x^{2}$

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			-	$x^{3}$	+	$4 x^{2}$

Step 2.1.5.5

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$

Step 2.1.5.6

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$33 x$

Step 2.1.5.7

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

				$x^{2}$	-	$8 x$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$33 x$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$33 x$
					-	$8 x^{2}$	+	$32 x$

Step 2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} + 32 x$

				$x^{2}$	-	$8 x$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$33 x$
					+	$8 x^{2}$	-	$32 x$

Step 2.1.5.10

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

				$x^{2}$	-	$8 x$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$33 x$
					+	$8 x^{2}$	-	$32 x$
							+	$x$

Step 2.1.5.11

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

				$x^{2}$	-	$8 x$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$33 x$
					+	$8 x^{2}$	-	$32 x$
							+	$x$	-	$4$

Step 2.1.5.12

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

				$x^{2}$	-	$8 x$	+	$1$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$33 x$
					+	$8 x^{2}$	-	$32 x$
							+	$x$	-	$4$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$	+	$1$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$33 x$
					+	$8 x^{2}$	-	$32 x$
							+	$x$	-	$4$
							+	$x$	-	$4$

Step 2.1.5.14

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

				$x^{2}$	-	$8 x$	+	$1$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$33 x$
					+	$8 x^{2}$	-	$32 x$
							+	$x$	-	$4$
							-	$x$	+	$4$

Step 2.1.5.15

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

				$x^{2}$	-	$8 x$	+	$1$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$33 x$	-	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$33 x$
					+	$8 x^{2}$	-	$32 x$
							+	$x$	-	$4$
							-	$x$	+	$4$
										$0$

Step 2.1.5.16

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

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

Step 2.1.6

Write $x^{3} - 12 x^{2} + 33 x - 4$ as a set of factors.

$(3 x - 1) ((x - 4) (x^{2} - 8 x + 1))$

Step 2.2

Remove unnecessary parentheses.

$(3 x - 1) (x - 4) (x^{2} - 8 x + 1)$