Finite Math Examples

$f (x) = 4 x^{3} - 19 x^{2} + 19 x + 6$

Step 1

Set $4 x^{3} - 19 x^{2} + 19 x + 6$ equal to $0$ .

$4 x^{3} - 19 x^{2} + 19 x + 6 = 0$

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 $4 x^{3} - 19 x^{2} + 19 x + 6$ 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.

$p = \pm 1, \pm 6, \pm 2, \pm 3$

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

Step 2.1.1.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 6, \pm 1.5, \pm 3, \pm 2, \pm 0.75$

Step 2.1.1.3

Substitute $- 0.25$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.25$ is a root of the polynomial.

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Step 2.1.1.3.1

Substitute $- 0.25$ into the polynomial.

$4 {(- 0.25)}^{3} - 19 {(- 0.25)}^{2} + 19 \cdot - 0.25 + 6$

Step 2.1.1.3.2

Raise $- 0.25$ to the power of $3$ .

$4 \cdot - 0.015625 - 19 {(- 0.25)}^{2} + 19 \cdot - 0.25 + 6$

Step 2.1.1.3.3

Multiply $4$ by $- 0.015625$ .

$- 0.0625 - 19 {(- 0.25)}^{2} + 19 \cdot - 0.25 + 6$

Step 2.1.1.3.4

Raise $- 0.25$ to the power of $2$ .

$- 0.0625 - 19 \cdot 0.0625 + 19 \cdot - 0.25 + 6$

Step 2.1.1.3.5

Multiply $- 19$ by $0.0625$ .

$- 0.0625 - 1.1875 + 19 \cdot - 0.25 + 6$

Step 2.1.1.3.6

Subtract $1.1875$ from $- 0.0625$ .

$- 1.25 + 19 \cdot - 0.25 + 6$

Step 2.1.1.3.7

Multiply $19$ by $- 0.25$ .

$- 1.25 - 4.75 + 6$

Step 2.1.1.3.8

Subtract $4.75$ from $- 1.25$ .

$- 6 + 6$

Step 2.1.1.3.9

Add $- 6$ and $6$ .

$0$

Step 2.1.1.4

Since $- 0.25$ is a known root, divide the polynomial by $4 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 19 x^{2} + 19 x + 6}{4 x + 1}$

Step 2.1.1.5

Divide $4 x^{3} - 19 x^{2} + 19 x + 6$ by $4 x + 1$ .

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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 $0$ .

$4 x$

$1$

$4 x^{3}$

$19 x^{2}$

$19 x$

$6$

Step 2.1.1.5.2

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

					$x^{2}$
	$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			+	$4 x^{3}$	+	$x^{2}$

Step 2.1.1.5.4

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

				$x^{2}$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			-	$4 x^{3}$	-	$x^{2}$

Step 2.1.1.5.5

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

				$x^{2}$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			-	$4 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$

Step 2.1.1.5.6

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

				$x^{2}$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			-	$4 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	+	$19 x$

Step 2.1.1.5.7

Divide the highest order term in the dividend $- 20 x^{2}$ by the highest order term in divisor $4 x$ .

				$x^{2}$	-	$5 x$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			-	$4 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	+	$19 x$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			-	$4 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	+	$19 x$
					-	$20 x^{2}$	-	$5 x$

Step 2.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x^{2} - 5 x$

				$x^{2}$	-	$5 x$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			-	$4 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	+	$19 x$
					+	$20 x^{2}$	+	$5 x$

Step 2.1.1.5.10

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

				$x^{2}$	-	$5 x$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			-	$4 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	+	$19 x$
					+	$20 x^{2}$	+	$5 x$
							+	$24 x$

Step 2.1.1.5.11

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

				$x^{2}$	-	$5 x$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			-	$4 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	+	$19 x$
					+	$20 x^{2}$	+	$5 x$
							+	$24 x$	+	$6$

Step 2.1.1.5.12

Divide the highest order term in the dividend $24 x$ by the highest order term in divisor $4 x$ .

				$x^{2}$	-	$5 x$	+	$6$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			-	$4 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	+	$19 x$
					+	$20 x^{2}$	+	$5 x$
							+	$24 x$	+	$6$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$	+	$6$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			-	$4 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	+	$19 x$
					+	$20 x^{2}$	+	$5 x$
							+	$24 x$	+	$6$
							+	$24 x$	+	$6$

Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $24 x + 6$

				$x^{2}$	-	$5 x$	+	$6$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			-	$4 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	+	$19 x$
					+	$20 x^{2}$	+	$5 x$
							+	$24 x$	+	$6$
							-	$24 x$	-	$6$

Step 2.1.1.5.15

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

				$x^{2}$	-	$5 x$	+	$6$
$4 x$	+	$1$		$4 x^{3}$	-	$19 x^{2}$	+	$19 x$	+	$6$
			-	$4 x^{3}$	-	$x^{2}$
					-	$20 x^{2}$	+	$19 x$
					+	$20 x^{2}$	+	$5 x$
							+	$24 x$	+	$6$
							-	$24 x$	-	$6$
										$0$

Step 2.1.1.5.16

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

$x^{2} - 5 x + 6$

Step 2.1.1.6

Write $4 x^{3} - 19 x^{2} + 19 x + 6$ as a set of factors.

$(4 x + 1) (x^{2} - 5 x + 6) = 0$

Step 2.1.2

Factor $x^{2} - 5 x + 6$ using the AC method.

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Step 2.1.2.1

Factor $x^{2} - 5 x + 6$ using the AC method.

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Step 2.1.2.1.1

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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 2.1.2.1.2

Write the factored form using these integers.

$(4 x + 1) ((x - 3) (x - 2)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$(4 x + 1) (x - 3) (x - 2) = 0$

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 $0$ .

$4 x + 1 = 0$

$x - 3 = 0$

$x - 2 = 0$

Step 2.3

Set $4 x + 1$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Step 2.3.2

Solve $4 x + 1 = 0$ for $x$ .

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Step 2.3.2.1

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Step 2.3.2.2

Divide each term in $4 x = - 1$ by $4$ and simplify.

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Step 2.3.2.2.1

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Step 2.3.2.2.3

Simplify the right side.

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Step 2.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

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 $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.6

The final solution is all the values that make $(4 x + 1) (x - 3) (x - 2) = 0$ true.

$x = - \frac{1}{4}, 3, 2$

Step 3