Finite Math Examples

$2 x^{3} - 5 x^{2} - 37 x + 60$

Step 1

Factor $2 x^{3} - 5 x^{2} - 37 x + 60$ 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.

$p = \pm 1, \pm 60, \pm 2, \pm 30, \pm 3, \pm 20, \pm 4, \pm 15, \pm 5, \pm 12, \pm 6, \pm 10$

$q = \pm 1, \pm 2$

Step 1.2

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

$\pm 1, \pm 0.5, \pm 60, \pm 30, \pm 2, \pm 15, \pm 3, \pm 1.5, \pm 20, \pm 10, \pm 4, \pm 7.5, \pm 5, \pm 2.5, \pm 12, \pm 6$

Step 1.3

Substitute $1.5$ and simplify the expression. In this case, the expression is equal to $0$ so $1.5$ is a root of the polynomial.

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Step 1.3.1

Substitute $1.5$ into the polynomial.

$2 \cdot {1.5}^{3} - 5 \cdot {1.5}^{2} - 37 \cdot 1.5 + 60$

Step 1.3.2

Raise $1.5$ to the power of $3$ .

$2 \cdot 3.375 - 5 \cdot {1.5}^{2} - 37 \cdot 1.5 + 60$

Step 1.3.3

Multiply $2$ by $3.375$ .

$6.75 - 5 \cdot {1.5}^{2} - 37 \cdot 1.5 + 60$

Step 1.3.4

Raise $1.5$ to the power of $2$ .

$6.75 - 5 \cdot 2.25 - 37 \cdot 1.5 + 60$

Step 1.3.5

Multiply $- 5$ by $2.25$ .

$6.75 - 11.25 - 37 \cdot 1.5 + 60$

Step 1.3.6

Subtract $11.25$ from $6.75$ .

$- 4.5 - 37 \cdot 1.5 + 60$

Step 1.3.7

Multiply $- 37$ by $1.5$ .

$- 4.5 - 55.5 + 60$

Step 1.3.8

Subtract $55.5$ from $- 4.5$ .

$- 60 + 60$

Step 1.3.9

Add $- 60$ and $60$ .

$0$

Step 1.4

Since $1.5$ is a known root, divide the polynomial by $2 x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} - 5 x^{2} - 37 x + 60}{2 x - 3}$

Step 1.5

Divide $2 x^{3} - 5 x^{2} - 37 x + 60$ by $2 x - 3$ .

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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 $0$ .

$2 x$

$3$

$2 x^{3}$

$5 x^{2}$

$37 x$

$60$

Step 1.5.2

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

					$x^{2}$
	$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			+	$2 x^{3}$	-	$3 x^{2}$

Step 1.5.4

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

				$x^{2}$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			-	$2 x^{3}$	+	$3 x^{2}$

Step 1.5.5

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

				$x^{2}$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			-	$2 x^{3}$	+	$3 x^{2}$
					-	$2 x^{2}$

Step 1.5.6

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

				$x^{2}$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			-	$2 x^{3}$	+	$3 x^{2}$
					-	$2 x^{2}$	-	$37 x$

Step 1.5.7

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

				$x^{2}$	-	$x$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			-	$2 x^{3}$	+	$3 x^{2}$
					-	$2 x^{2}$	-	$37 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			-	$2 x^{3}$	+	$3 x^{2}$
					-	$2 x^{2}$	-	$37 x$
					-	$2 x^{2}$	+	$3 x$

Step 1.5.9

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

				$x^{2}$	-	$x$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			-	$2 x^{3}$	+	$3 x^{2}$
					-	$2 x^{2}$	-	$37 x$
					+	$2 x^{2}$	-	$3 x$

Step 1.5.10

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

				$x^{2}$	-	$x$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			-	$2 x^{3}$	+	$3 x^{2}$
					-	$2 x^{2}$	-	$37 x$
					+	$2 x^{2}$	-	$3 x$
							-	$40 x$

Step 1.5.11

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

				$x^{2}$	-	$x$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			-	$2 x^{3}$	+	$3 x^{2}$
					-	$2 x^{2}$	-	$37 x$
					+	$2 x^{2}$	-	$3 x$
							-	$40 x$	+	$60$

Step 1.5.12

Divide the highest order term in the dividend $- 40 x$ by the highest order term in divisor $2 x$ .

				$x^{2}$	-	$x$	-	$20$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			-	$2 x^{3}$	+	$3 x^{2}$
					-	$2 x^{2}$	-	$37 x$
					+	$2 x^{2}$	-	$3 x$
							-	$40 x$	+	$60$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	-	$20$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			-	$2 x^{3}$	+	$3 x^{2}$
					-	$2 x^{2}$	-	$37 x$
					+	$2 x^{2}$	-	$3 x$
							-	$40 x$	+	$60$
							-	$40 x$	+	$60$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 40 x + 60$

				$x^{2}$	-	$x$	-	$20$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			-	$2 x^{3}$	+	$3 x^{2}$
					-	$2 x^{2}$	-	$37 x$
					+	$2 x^{2}$	-	$3 x$
							-	$40 x$	+	$60$
							+	$40 x$	-	$60$

Step 1.5.15

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

				$x^{2}$	-	$x$	-	$20$
$2 x$	-	$3$		$2 x^{3}$	-	$5 x^{2}$	-	$37 x$	+	$60$
			-	$2 x^{3}$	+	$3 x^{2}$
					-	$2 x^{2}$	-	$37 x$
					+	$2 x^{2}$	-	$3 x$
							-	$40 x$	+	$60$
							+	$40 x$	-	$60$
										$0$

Step 1.5.16

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

$x^{2} - x - 20$

Step 1.6

Write $2 x^{3} - 5 x^{2} - 37 x + 60$ as a set of factors.

$(2 x - 3) (x^{2} - x - 20)$

Step 2

Factor $x^{2} - x - 20$ using the AC method.

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Step 2.1

Factor $x^{2} - x - 20$ using the AC method.

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Step 2.1.1

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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 2.1.2

Write the factored form using these integers.

$(2 x - 3) ((x - 5) (x + 4))$

Step 2.2

Remove unnecessary parentheses.

$(2 x - 3) (x - 5) (x + 4)$