Finite Math Examples

$42 x^{4} + 335 x^{3} + 663 x^{2} - 24 x - 16$

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.

$p = \pm 1, \pm 2, \pm 4, \pm 8, \pm 16$

$q = \pm 1, \pm 2, \pm 3, \pm 6, \pm 7, \pm 14, \pm 21, \pm 42$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm \frac{1}{7}, \pm \frac{1}{14}, \pm \frac{1}{21}, \pm \frac{1}{42}, \pm 2, \pm \frac{2}{3}, \pm \frac{2}{7}, \pm \frac{2}{21}, \pm 4, \pm \frac{4}{3}, \pm \frac{4}{7}, \pm \frac{4}{21}, \pm 8, \pm \frac{8}{3}, \pm \frac{8}{7}, \pm \frac{8}{21}, \pm 16, \pm \frac{16}{3}, \pm \frac{16}{7}, \pm \frac{16}{21}$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

$42 {(- 4)}^{4} + 335 {(- 4)}^{3} + 663 {(- 4)}^{2} - 24 \cdot - 4 - 16$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 4$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

$42 \cdot 256 + 335 {(- 4)}^{3} + 663 {(- 4)}^{2} - 24 \cdot - 4 - 16$

Step 4.1.2

Multiply $42$ by $256$ .

$10752 + 335 {(- 4)}^{3} + 663 {(- 4)}^{2} - 24 \cdot - 4 - 16$

Step 4.1.3

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

$10752 + 335 \cdot - 64 + 663 {(- 4)}^{2} - 24 \cdot - 4 - 16$

Step 4.1.4

Multiply $335$ by $- 64$ .

$10752 - 21440 + 663 {(- 4)}^{2} - 24 \cdot - 4 - 16$

Step 4.1.5

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

$10752 - 21440 + 663 \cdot 16 - 24 \cdot - 4 - 16$

Step 4.1.6

Multiply $663$ by $16$ .

$10752 - 21440 + 10608 - 24 \cdot - 4 - 16$

Step 4.1.7

Multiply $- 24$ by $- 4$ .

$10752 - 21440 + 10608 + 96 - 16$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $21440$ from $10752$ .

$- 10688 + 10608 + 96 - 16$

Step 4.2.2

Add $- 10688$ and $10608$ .

$- 80 + 96 - 16$

Step 4.2.3

Add $- 80$ and $96$ .

$16 - 16$

Step 4.2.4

Subtract $16$ from $16$ .

$0$

Step 5

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.

$\frac{42 x^{4} + 335 x^{3} + 663 x^{2} - 24 x - 16}{x + 4}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Step 6.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 4$	$42$	$335$	$663$	$- 24$	$- 16$

Step 6.2

The first number in the dividend $(42)$ is put into the first position of the result area (below the horizontal line).

$- 4$	$42$	$335$	$663$	$- 24$	$- 16$

	$42$

Step 6.3

Multiply the newest entry in the result $(42)$ by the divisor $(- 4)$ and place the result of $(- 168)$ under the next term in the dividend $(335)$ .

$- 4$	$42$	$335$	$663$	$- 24$	$- 16$
		$- 168$
	$42$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 4$	$42$	$335$	$663$	$- 24$	$- 16$
		$- 168$
	$42$	$167$

Step 6.5

Multiply the newest entry in the result $(167)$ by the divisor $(- 4)$ and place the result of $(- 668)$ under the next term in the dividend $(663)$ .

$- 4$	$42$	$335$	$663$	$- 24$	$- 16$
		$- 168$	$- 668$
	$42$	$167$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 4$	$42$	$335$	$663$	$- 24$	$- 16$
		$- 168$	$- 668$
	$42$	$167$	$- 5$

Step 6.7

Multiply the newest entry in the result $(- 5)$ by the divisor $(- 4)$ and place the result of $(20)$ under the next term in the dividend $(- 24)$ .

$- 4$	$42$	$335$	$663$	$- 24$	$- 16$
		$- 168$	$- 668$	$20$
	$42$	$167$	$- 5$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 4$	$42$	$335$	$663$	$- 24$	$- 16$
		$- 168$	$- 668$	$20$
	$42$	$167$	$- 5$	$- 4$

Step 6.9

Multiply the newest entry in the result $(- 4)$ by the divisor $(- 4)$ and place the result of $(16)$ under the next term in the dividend $(- 16)$ .

$- 4$	$42$	$335$	$663$	$- 24$	$- 16$
		$- 168$	$- 668$	$20$	$16$
	$42$	$167$	$- 5$	$- 4$

Step 6.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 4$	$42$	$335$	$663$	$- 24$	$- 16$
		$- 168$	$- 668$	$20$	$16$
	$42$	$167$	$- 5$	$- 4$	$0$

Step 6.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$42 x^{3} + 167 x^{2} + (- 5) x - 4$

Step 6.12

Simplify the quotient polynomial.

$42 x^{3} + 167 x^{2} - 5 x - 4$

Step 7

Solve the equation to find any remaining roots.

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Step 7.1

Factor the left side of the equation.

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Step 7.1.1

Factor $42 x^{3} + 167 x^{2} - 5 x - 4$ using the rational roots test.

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Step 7.1.1.1

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.

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

$q = \pm 1, \pm 42, \pm 2, \pm 21, \pm 3, \pm 14, \pm 6, \pm 7$

Step 7.1.1.2

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

$\pm 1, \pm 0.0\overline{238095}, \pm 0.5, \pm 0.\overline{047619}, \pm 0.\overline{3}, \pm 0.0\overline{714285}, \pm 0.1\overline{6}, \pm 0.\overline{142857}, \pm 4, \pm 0.\overline{095238}, \pm 2, \pm 0.\overline{190476}, \pm 1.\overline{3}, \pm 0.\overline{285714}, \pm 0.\overline{6}, \pm 0.\overline{571428}$

Step 7.1.1.3

Substitute $- 0.\overline{142857}$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.\overline{142857}$ is a root of the polynomial.

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Step 7.1.1.3.1

Substitute $- 0.\overline{142857}$ into the polynomial.

$42 {(- 0.\overline{142857})}^{3} + 167 {(- 0.\overline{142857})}^{2} - 5 \cdot - 0.\overline{142857} - 4$

Step 7.1.1.3.2

Raise $- 0.\overline{142857}$ to the power of $3$ .

$42 \cdot - 0.00291545 + 167 {(- 0.\overline{142857})}^{2} - 5 \cdot - 0.\overline{142857} - 4$

Step 7.1.1.3.3

Multiply $42$ by $- 0.00291545$ .

$- 0.12244897 + 167 {(- 0.\overline{142857})}^{2} - 5 \cdot - 0.\overline{142857} - 4$

Step 7.1.1.3.4

Raise $- 0.\overline{142857}$ to the power of $2$ .

$- 0.12244897 + 167 \cdot 0.02040816 - 5 \cdot - 0.\overline{142857} - 4$

Step 7.1.1.3.5

Multiply $167$ by $0.02040816$ .

$- 0.12244897 + 3.40816326 - 5 \cdot - 0.\overline{142857} - 4$

Step 7.1.1.3.6

Add $- 0.12244897$ and $3.40816326$ .

$3.28571428 - 5 \cdot - 0.\overline{142857} - 4$

Step 7.1.1.3.7

Multiply $- 5$ by $- 0.\overline{142857}$ .

$3.28571428 + 0.\overline{714285} - 4$

Step 7.1.1.3.8

Add $3.28571428$ and $0.\overline{714285}$ .

$4 - 4$

Step 7.1.1.3.9

Subtract $4$ from $4$ .

$0$

Step 7.1.1.4

Since $- 0.\overline{142857}$ is a known root, divide the polynomial by $7 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{42 x^{3} + 167 x^{2} - 5 x - 4}{7 x + 1}$

Step 7.1.1.5

Divide $42 x^{3} + 167 x^{2} - 5 x - 4$ by $7 x + 1$ .

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Step 7.1.1.5.1

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

$7 x$

$1$

$42 x^{3}$

$167 x^{2}$

$5 x$

$4$

Step 7.1.1.5.2

Divide the highest order term in the dividend $42 x^{3}$ by the highest order term in divisor $7 x$ .

					$6 x^{2}$
	$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$

Step 7.1.1.5.3

Multiply the new quotient term by the divisor.

				$6 x^{2}$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			+	$42 x^{3}$	+	$6 x^{2}$

Step 7.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $42 x^{3} + 6 x^{2}$

				$6 x^{2}$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			-	$42 x^{3}$	-	$6 x^{2}$

Step 7.1.1.5.5

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

				$6 x^{2}$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			-	$42 x^{3}$	-	$6 x^{2}$
					+	$161 x^{2}$

Step 7.1.1.5.6

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

				$6 x^{2}$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			-	$42 x^{3}$	-	$6 x^{2}$
					+	$161 x^{2}$	-	$5 x$

Step 7.1.1.5.7

Divide the highest order term in the dividend $161 x^{2}$ by the highest order term in divisor $7 x$ .

				$6 x^{2}$	+	$23 x$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			-	$42 x^{3}$	-	$6 x^{2}$
					+	$161 x^{2}$	-	$5 x$

Step 7.1.1.5.8

Multiply the new quotient term by the divisor.

				$6 x^{2}$	+	$23 x$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			-	$42 x^{3}$	-	$6 x^{2}$
					+	$161 x^{2}$	-	$5 x$
					+	$161 x^{2}$	+	$23 x$

Step 7.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $161 x^{2} + 23 x$

				$6 x^{2}$	+	$23 x$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			-	$42 x^{3}$	-	$6 x^{2}$
					+	$161 x^{2}$	-	$5 x$
					-	$161 x^{2}$	-	$23 x$

Step 7.1.1.5.10

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

				$6 x^{2}$	+	$23 x$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			-	$42 x^{3}$	-	$6 x^{2}$
					+	$161 x^{2}$	-	$5 x$
					-	$161 x^{2}$	-	$23 x$
							-	$28 x$

Step 7.1.1.5.11

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

				$6 x^{2}$	+	$23 x$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			-	$42 x^{3}$	-	$6 x^{2}$
					+	$161 x^{2}$	-	$5 x$
					-	$161 x^{2}$	-	$23 x$
							-	$28 x$	-	$4$

Step 7.1.1.5.12

Divide the highest order term in the dividend $- 28 x$ by the highest order term in divisor $7 x$ .

				$6 x^{2}$	+	$23 x$	-	$4$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			-	$42 x^{3}$	-	$6 x^{2}$
					+	$161 x^{2}$	-	$5 x$
					-	$161 x^{2}$	-	$23 x$
							-	$28 x$	-	$4$

Step 7.1.1.5.13

Multiply the new quotient term by the divisor.

				$6 x^{2}$	+	$23 x$	-	$4$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			-	$42 x^{3}$	-	$6 x^{2}$
					+	$161 x^{2}$	-	$5 x$
					-	$161 x^{2}$	-	$23 x$
							-	$28 x$	-	$4$
							-	$28 x$	-	$4$

Step 7.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 28 x - 4$

				$6 x^{2}$	+	$23 x$	-	$4$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			-	$42 x^{3}$	-	$6 x^{2}$
					+	$161 x^{2}$	-	$5 x$
					-	$161 x^{2}$	-	$23 x$
							-	$28 x$	-	$4$
							+	$28 x$	+	$4$

Step 7.1.1.5.15

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

				$6 x^{2}$	+	$23 x$	-	$4$
$7 x$	+	$1$		$42 x^{3}$	+	$167 x^{2}$	-	$5 x$	-	$4$
			-	$42 x^{3}$	-	$6 x^{2}$
					+	$161 x^{2}$	-	$5 x$
					-	$161 x^{2}$	-	$23 x$
							-	$28 x$	-	$4$
							+	$28 x$	+	$4$
										$0$

Step 7.1.1.5.16

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

$6 x^{2} + 23 x - 4$

Step 7.1.1.6

Write $42 x^{3} + 167 x^{2} - 5 x - 4$ as a set of factors.

$(7 x + 1) (6 x^{2} + 23 x - 4) = 0$

Step 7.1.2

Factor by grouping.

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Step 7.1.2.1

Factor by grouping.

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Step 7.1.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 6 \cdot - 4 = - 24$ and whose sum is $b = 23$ .

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Step 7.1.2.1.1.1

Factor $23$ out of $23 x$ .

$(7 x + 1) (6 x^{2} + 23 (x) - 4) = 0$

Step 7.1.2.1.1.2

Rewrite $23$ as $- 1$ plus $24$

$(7 x + 1) (6 x^{2} + (- 1 + 24) x - 4) = 0$

Step 7.1.2.1.1.3

Apply the distributive property.

$(7 x + 1) (6 x^{2} - 1 x + 24 x - 4) = 0$

Step 7.1.2.1.2

Factor out the greatest common factor from each group.

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Step 7.1.2.1.2.1

Group the first two terms and the last two terms.

$(7 x + 1) ((6 x^{2} - 1 x) + 24 x - 4) = 0$

Step 7.1.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$(7 x + 1) (x (6 x - 1) + 4 (6 x - 1)) = 0$

Step 7.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $6 x - 1$ .

$(7 x + 1) ((6 x - 1) (x + 4)) = 0$

Step 7.1.2.2

Remove unnecessary parentheses.

$(7 x + 1) (6 x - 1) (x + 4) = 0$

Step 7.2

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

$7 x + 1 = 0$

$6 x - 1 = 0$

$x + 4 = 0$

Step 7.3

Set $7 x + 1$ equal to $0$ and solve for $x$ .

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Step 7.3.1

Set $7 x + 1$ equal to $0$ .

$7 x + 1 = 0$

Step 7.3.2

Solve $7 x + 1 = 0$ for $x$ .

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Step 7.3.2.1

Subtract $1$ from both sides of the equation.

$7 x = - 1$

Step 7.3.2.2

Divide each term in $7 x = - 1$ by $7$ and simplify.

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Step 7.3.2.2.1

Divide each term in $7 x = - 1$ by $7$ .

$\frac{7 x}{7} = \frac{- 1}{7}$

Step 7.3.2.2.2

Simplify the left side.

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Step 7.3.2.2.2.1

Cancel the common factor of $7$ .

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Step 7.3.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 1}{7}$

Step 7.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{7}$

Step 7.3.2.2.3

Simplify the right side.

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Step 7.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{7}$

Step 7.4

Set $6 x - 1$ equal to $0$ and solve for $x$ .

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Step 7.4.1

Set $6 x - 1$ equal to $0$ .

$6 x - 1 = 0$

Step 7.4.2

Solve $6 x - 1 = 0$ for $x$ .

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Step 7.4.2.1

Add $1$ to both sides of the equation.

$6 x = 1$

Step 7.4.2.2

Divide each term in $6 x = 1$ by $6$ and simplify.

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Step 7.4.2.2.1

Divide each term in $6 x = 1$ by $6$ .

$\frac{6 x}{6} = \frac{1}{6}$

Step 7.4.2.2.2

Simplify the left side.

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Step 7.4.2.2.2.1

Cancel the common factor of $6$ .

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Step 7.4.2.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{1}{6}$

Step 7.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{6}$

Step 7.5

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 7.5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 7.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7.6

The final solution is all the values that make $(7 x + 1) (6 x - 1) (x + 4) = 0$ true.

$x = - \frac{1}{7}, \frac{1}{6}, - 4$

Step 8

The polynomial can be written as a set of linear factors.

$(x + 4) (x + \frac{1}{7}) (x - \frac{1}{6})$

Step 9

These are the roots (zeros) of the polynomial $42 x^{4} + 335 x^{3} + 663 x^{2} - 24 x - 16$ .

$x = - 4, - \frac{1}{7}, \frac{1}{6}$

Step 10