Finite Math Examples

$y = x^{3} + 9 x^{2} + 15 x - 25$

Step 1

Set $x^{3} + 9 x^{2} + 15 x - 25$ equal to $0$ .

$x^{3} + 9 x^{2} + 15 x - 25 = 0$

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 $x^{3} + 9 x^{2} + 15 x - 25$ 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.

$p = \pm 1, \pm 25, \pm 5$

$q = \pm 1$

Step 2.1.1.2

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

$\pm 1, \pm 25, \pm 5$

Step 2.1.1.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 2.1.1.3.1

Substitute $1$ into the polynomial.

$1^{3} + 9 \cdot 1^{2} + 15 \cdot 1 - 25$

Step 2.1.1.3.2

Raise $1$ to the power of $3$ .

$1 + 9 \cdot 1^{2} + 15 \cdot 1 - 25$

Step 2.1.1.3.3

Raise $1$ to the power of $2$ .

$1 + 9 \cdot 1 + 15 \cdot 1 - 25$

Step 2.1.1.3.4

Multiply $9$ by $1$ .

$1 + 9 + 15 \cdot 1 - 25$

Step 2.1.1.3.5

Add $1$ and $9$ .

$10 + 15 \cdot 1 - 25$

Step 2.1.1.3.6

Multiply $15$ by $1$ .

$10 + 15 - 25$

Step 2.1.1.3.7

Add $10$ and $15$ .

$25 - 25$

Step 2.1.1.3.8

Subtract $25$ from $25$ .

$0$

Step 2.1.1.4

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

$\frac{x^{3} + 9 x^{2} + 15 x - 25}{x - 1}$

Step 2.1.1.5

Divide $x^{3} + 9 x^{2} + 15 x - 25$ by $x - 1$ .

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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 $0$ .

$x$

$1$

$x^{3}$

$9 x^{2}$

$15 x$

$25$

Step 2.1.1.5.2

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			+	$x^{3}$	-	$x^{2}$

Step 2.1.1.5.4

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			-	$x^{3}$	+	$x^{2}$

Step 2.1.1.5.5

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			-	$x^{3}$	+	$x^{2}$
					+	$10 x^{2}$

Step 2.1.1.5.6

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			-	$x^{3}$	+	$x^{2}$
					+	$10 x^{2}$	+	$15 x$

Step 2.1.1.5.7

Divide the highest order term in the dividend $10 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$10 x$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			-	$x^{3}$	+	$x^{2}$
					+	$10 x^{2}$	+	$15 x$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$10 x$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			-	$x^{3}$	+	$x^{2}$
					+	$10 x^{2}$	+	$15 x$
					+	$10 x^{2}$	-	$10 x$

Step 2.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{2} - 10 x$

				$x^{2}$	+	$10 x$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			-	$x^{3}$	+	$x^{2}$
					+	$10 x^{2}$	+	$15 x$
					-	$10 x^{2}$	+	$10 x$

Step 2.1.1.5.10

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

				$x^{2}$	+	$10 x$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			-	$x^{3}$	+	$x^{2}$
					+	$10 x^{2}$	+	$15 x$
					-	$10 x^{2}$	+	$10 x$
							+	$25 x$

Step 2.1.1.5.11

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

				$x^{2}$	+	$10 x$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			-	$x^{3}$	+	$x^{2}$
					+	$10 x^{2}$	+	$15 x$
					-	$10 x^{2}$	+	$10 x$
							+	$25 x$	-	$25$

Step 2.1.1.5.12

Divide the highest order term in the dividend $25 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$10 x$	+	$25$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			-	$x^{3}$	+	$x^{2}$
					+	$10 x^{2}$	+	$15 x$
					-	$10 x^{2}$	+	$10 x$
							+	$25 x$	-	$25$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$10 x$	+	$25$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			-	$x^{3}$	+	$x^{2}$
					+	$10 x^{2}$	+	$15 x$
					-	$10 x^{2}$	+	$10 x$
							+	$25 x$	-	$25$
							+	$25 x$	-	$25$

Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $25 x - 25$

				$x^{2}$	+	$10 x$	+	$25$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			-	$x^{3}$	+	$x^{2}$
					+	$10 x^{2}$	+	$15 x$
					-	$10 x^{2}$	+	$10 x$
							+	$25 x$	-	$25$
							-	$25 x$	+	$25$

Step 2.1.1.5.15

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

				$x^{2}$	+	$10 x$	+	$25$
$x$	-	$1$		$x^{3}$	+	$9 x^{2}$	+	$15 x$	-	$25$
			-	$x^{3}$	+	$x^{2}$
					+	$10 x^{2}$	+	$15 x$
					-	$10 x^{2}$	+	$10 x$
							+	$25 x$	-	$25$
							-	$25 x$	+	$25$
										$0$

Step 2.1.1.5.16

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

$x^{2} + 10 x + 25$

Step 2.1.1.6

Write $x^{3} + 9 x^{2} + 15 x - 25$ as a set of factors.

$(x - 1) (x^{2} + 10 x + 25) = 0$

Step 2.1.2

Factor using the perfect square rule.

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Step 2.1.2.1

Rewrite $25$ as $5^{2}$ .

$(x - 1) (x^{2} + 10 x + 5^{2}) = 0$

Step 2.1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 2.1.2.3

Rewrite the polynomial.

$(x - 1) (x^{2} + 2 \cdot x \cdot 5 + 5^{2}) = 0$

Step 2.1.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

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

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 $0$ .

$x - 1 = 0$

${(x + 5)}^{2} = 0$

Step 2.3

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

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Step 2.3.1

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

$x - 1 = 0$

Step 2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.4

Set ${(x + 5)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set ${(x + 5)}^{2}$ equal to $0$ .

${(x + 5)}^{2} = 0$

Step 2.4.2

Solve ${(x + 5)}^{2} = 0$ for $x$ .

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Step 2.4.2.1

Set the $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 2.4.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 2.5

The final solution is all the values that make $(x - 1) {(x + 5)}^{2} = 0$ true.

$x = 1, - 5$

Step 3