Finite Math Examples

$f (x) = 10 x^{4} - 3 x^{3} - 63 x^{2} + 152 x - 188$

Step 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.

$p = \pm 1, \pm 188, \pm 2, \pm 94, \pm 4, \pm 47$

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

Step 2

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

$\pm 1, \pm 0.1, \pm 0.5, \pm 0.2, \pm 188, \pm 18.8, \pm 94, \pm 37.6, \pm 2, \pm 0.4, \pm 9.4, \pm 47, \pm 4, \pm 0.8, \pm 4.7, \pm 23.5$

Step 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 3.1

Substitute $2$ into the polynomial.

$10 \cdot 2^{4} - 3 \cdot 2^{3} - 63 \cdot 2^{2} + 152 \cdot 2 - 188$

Step 3.2

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

$10 \cdot 16 - 3 \cdot 2^{3} - 63 \cdot 2^{2} + 152 \cdot 2 - 188$

Step 3.3

Multiply $10$ by $16$ .

$160 - 3 \cdot 2^{3} - 63 \cdot 2^{2} + 152 \cdot 2 - 188$

Step 3.4

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

$160 - 3 \cdot 8 - 63 \cdot 2^{2} + 152 \cdot 2 - 188$

Step 3.5

Multiply $- 3$ by $8$ .

$160 - 24 - 63 \cdot 2^{2} + 152 \cdot 2 - 188$

Step 3.6

Subtract $24$ from $160$ .

$136 - 63 \cdot 2^{2} + 152 \cdot 2 - 188$

Step 3.7

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

$136 - 63 \cdot 4 + 152 \cdot 2 - 188$

Step 3.8

Multiply $- 63$ by $4$ .

$136 - 252 + 152 \cdot 2 - 188$

Step 3.9

Subtract $252$ from $136$ .

$- 116 + 152 \cdot 2 - 188$

Step 3.10

Multiply $152$ by $2$ .

$- 116 + 304 - 188$

Step 3.11

Add $- 116$ and $304$ .

$188 - 188$

Step 3.12

Subtract $188$ from $188$ .

$0$

Step 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.

$\frac{10 x^{4} - 3 x^{3} - 63 x^{2} + 152 x - 188}{x - 2}$

Step 5

Divide $10 x^{4} - 3 x^{3} - 63 x^{2} + 152 x - 188$ by $x - 2$ .

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Step 5.1

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

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

Step 5.2

Divide the highest order term in the dividend $10 x^{4}$ by the highest order term in divisor $x$ .

					$10 x^{3}$
	$x$	-	$2$		$10 x^{4}$	-	$3 x^{3}$	-	$63 x^{2}$	+	$152 x$	-	$188$

Step 5.3

Multiply the new quotient term by the divisor.

				$10 x^{3}$
$x$	-	$2$		$10 x^{4}$	-	$3 x^{3}$	-	$63 x^{2}$	+	$152 x$	-	$188$
			+	$10 x^{4}$	-	$20 x^{3}$

Step 5.4

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{4} - 20 x^{3}$

				$10 x^{3}$
$x$	-	$2$		$10 x^{4}$	-	$3 x^{3}$	-	$63 x^{2}$	+	$152 x$	-	$188$
			-	$10 x^{4}$	+	$20 x^{3}$

Step 5.5

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

$10 x^{3}$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

Step 5.6

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

$10 x^{3}$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

Step 5.7

Divide the highest order term in the dividend $17 x^{3}$ by the highest order term in divisor $x$ .

$10 x^{3}$

$17 x^{2}$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

Step 5.8

Multiply the new quotient term by the divisor.

$10 x^{3}$

$17 x^{2}$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

Step 5.9

The expression needs to be subtracted from the dividend, so change all the signs in $17 x^{3} - 34 x^{2}$

$10 x^{3}$

$17 x^{2}$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

Step 5.10

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

$10 x^{3}$

$17 x^{2}$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

$29 x^{2}$

Step 5.11

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

$10 x^{3}$

$17 x^{2}$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

$29 x^{2}$

$152 x$

Step 5.12

Divide the highest order term in the dividend $- 29 x^{2}$ by the highest order term in divisor $x$ .

$10 x^{3}$

$17 x^{2}$

$29 x$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

$29 x^{2}$

$152 x$

Step 5.13

Multiply the new quotient term by the divisor.

$10 x^{3}$

$17 x^{2}$

$29 x$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

$29 x^{2}$

$152 x$

$29 x^{2}$

$58 x$

Step 5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 29 x^{2} + 58 x$

$10 x^{3}$

$17 x^{2}$

$29 x$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

$29 x^{2}$

$152 x$

$29 x^{2}$

$58 x$

Step 5.15

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

$10 x^{3}$

$17 x^{2}$

$29 x$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

$29 x^{2}$

$152 x$

$29 x^{2}$

$58 x$

$94 x$

Step 5.16

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

$10 x^{3}$

$17 x^{2}$

$29 x$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

$29 x^{2}$

$152 x$

$29 x^{2}$

$58 x$

$94 x$

$188$

Step 5.17

Divide the highest order term in the dividend $94 x$ by the highest order term in divisor $x$ .

$10 x^{3}$

$17 x^{2}$

$29 x$

$94$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

$29 x^{2}$

$152 x$

$29 x^{2}$

$58 x$

$94 x$

$188$

Step 5.18

Multiply the new quotient term by the divisor.

$10 x^{3}$

$17 x^{2}$

$29 x$

$94$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

$29 x^{2}$

$152 x$

$29 x^{2}$

$58 x$

$94 x$

$188$

$94 x$

$188$

Step 5.19

The expression needs to be subtracted from the dividend, so change all the signs in $94 x - 188$

$10 x^{3}$

$17 x^{2}$

$29 x$

$94$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

$29 x^{2}$

$152 x$

$29 x^{2}$

$58 x$

$94 x$

$188$

$94 x$

$188$

Step 5.20

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

$10 x^{3}$

$17 x^{2}$

$29 x$

$94$

$x$

$2$

$10 x^{4}$

$3 x^{3}$

$63 x^{2}$

$152 x$

$188$

$10 x^{4}$

$20 x^{3}$

$17 x^{3}$

$63 x^{2}$

$17 x^{3}$

$34 x^{2}$

$29 x^{2}$

$152 x$

$29 x^{2}$

$58 x$

$94 x$

$188$

$94 x$

$188$

$0$

Step 5.21

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

$10 x^{3} + 17 x^{2} - 29 x + 94$

Step 6

Write $10 x^{4} - 3 x^{3} - 63 x^{2} + 152 x - 188$ as a set of factors.

$f (x) = (x - 2) (10 x^{3} + 17 x^{2} - 29 x + 94)$