Finite Math Examples

$f (x) = 4 x^{3} - 25 x^{2} - 154 x + 40$

Step 1

Set $4 x^{3} - 25 x^{2} - 154 x + 40$ equal to $0$ .

$4 x^{3} - 25 x^{2} - 154 x + 40 = 0$

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 $4 x^{3} - 25 x^{2} - 154 x + 40$ 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.

$p = \pm 1, \pm 40, \pm 2, \pm 20, \pm 4, \pm 10, \pm 5, \pm 8$

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

Step 2.1.1.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 40, \pm 10, \pm 20, \pm 2, \pm 5, \pm 4, \pm 2.5, \pm 1.25, \pm 8$

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.

$4 \cdot {0.25}^{3} - 25 \cdot {0.25}^{2} - 154 \cdot 0.25 + 40$

Step 2.1.1.3.2

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

$4 \cdot 0.015625 - 25 \cdot {0.25}^{2} - 154 \cdot 0.25 + 40$

Step 2.1.1.3.3

Multiply $4$ by $0.015625$ .

$0.0625 - 25 \cdot {0.25}^{2} - 154 \cdot 0.25 + 40$

Step 2.1.1.3.4

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

$0.0625 - 25 \cdot 0.0625 - 154 \cdot 0.25 + 40$

Step 2.1.1.3.5

Multiply $- 25$ by $0.0625$ .

$0.0625 - 1.5625 - 154 \cdot 0.25 + 40$

Step 2.1.1.3.6

Subtract $1.5625$ from $0.0625$ .

$- 1.5 - 154 \cdot 0.25 + 40$

Step 2.1.1.3.7

Multiply $- 154$ by $0.25$ .

$- 1.5 - 38.5 + 40$

Step 2.1.1.3.8

Subtract $38.5$ from $- 1.5$ .

$- 40 + 40$

Step 2.1.1.3.9

Add $- 40$ and $40$ .

$0$

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.

$\frac{4 x^{3} - 25 x^{2} - 154 x + 40}{4 x - 1}$

Step 2.1.1.5

Divide $4 x^{3} - 25 x^{2} - 154 x + 40$ 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 $0$ .

$4 x$

$1$

$4 x^{3}$

$25 x^{2}$

$154 x$

$40$

Step 2.1.1.5.2

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

					$x^{2}$
	$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			+	$4 x^{3}$	-	$x^{2}$

Step 2.1.1.5.4

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

				$x^{2}$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			-	$4 x^{3}$	+	$x^{2}$

Step 2.1.1.5.5

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

				$x^{2}$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			-	$4 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$

Step 2.1.1.5.6

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

				$x^{2}$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			-	$4 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	-	$154 x$

Step 2.1.1.5.7

Divide the highest order term in the dividend $- 24 x^{2}$ by the highest order term in divisor $4 x$ .

				$x^{2}$	-	$6 x$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			-	$4 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	-	$154 x$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			-	$4 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	-	$154 x$
					-	$24 x^{2}$	+	$6 x$

Step 2.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x^{2} + 6 x$

				$x^{2}$	-	$6 x$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			-	$4 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	-	$154 x$
					+	$24 x^{2}$	-	$6 x$

Step 2.1.1.5.10

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

				$x^{2}$	-	$6 x$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			-	$4 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	-	$154 x$
					+	$24 x^{2}$	-	$6 x$
							-	$160 x$

Step 2.1.1.5.11

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

				$x^{2}$	-	$6 x$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			-	$4 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	-	$154 x$
					+	$24 x^{2}$	-	$6 x$
							-	$160 x$	+	$40$

Step 2.1.1.5.12

Divide the highest order term in the dividend $- 160 x$ by the highest order term in divisor $4 x$ .

				$x^{2}$	-	$6 x$	-	$40$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			-	$4 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	-	$154 x$
					+	$24 x^{2}$	-	$6 x$
							-	$160 x$	+	$40$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	-	$40$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			-	$4 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	-	$154 x$
					+	$24 x^{2}$	-	$6 x$
							-	$160 x$	+	$40$
							-	$160 x$	+	$40$

Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 160 x + 40$

				$x^{2}$	-	$6 x$	-	$40$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			-	$4 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	-	$154 x$
					+	$24 x^{2}$	-	$6 x$
							-	$160 x$	+	$40$
							+	$160 x$	-	$40$

Step 2.1.1.5.15

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

				$x^{2}$	-	$6 x$	-	$40$
$4 x$	-	$1$		$4 x^{3}$	-	$25 x^{2}$	-	$154 x$	+	$40$
			-	$4 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	-	$154 x$
					+	$24 x^{2}$	-	$6 x$
							-	$160 x$	+	$40$
							+	$160 x$	-	$40$
										$0$

Step 2.1.1.5.16

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

$x^{2} - 6 x - 40$

Step 2.1.1.6

Write $4 x^{3} - 25 x^{2} - 154 x + 40$ as a set of factors.

$(4 x - 1) (x^{2} - 6 x - 40) = 0$

Step 2.1.2

Factor $x^{2} - 6 x - 40$ using the AC method.

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Step 2.1.2.1

Factor $x^{2} - 6 x - 40$ 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 $- 40$ and whose sum is $- 6$ .

$- 10, 4$

Step 2.1.2.1.2

Write the factored form using these integers.

$(4 x - 1) ((x - 10) (x + 4)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$(4 x - 1) (x - 10) (x + 4) = 0$

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 $0$ .

$4 x - 1 = 0$

$x - 10 = 0$

$x + 4 = 0$

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

Add $1$ to 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 2.4

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 2.4.2

Add $10$ to both sides of the equation.

$x = 10$

Step 2.5

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

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Step 2.5.1

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

$x + 4 = 0$

Step 2.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.6

The final solution is all the values that make $(4 x - 1) (x - 10) (x + 4) = 0$ true.

$x = \frac{1}{4}, 10, - 4$

Step 3