Finite Math Examples

$3 x^{3} + 10 x^{2} + x - 6 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $3 x^{3} + 10 x^{2} + x - 6$ using the rational roots test.

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Step 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.

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

$q = \pm 1, \pm 3$

Step 1.1.2

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

$\pm 1, \pm 0.\overline{3}, \pm 6, \pm 2, \pm 0.\overline{6}, \pm 3$

Step 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 1.1.3.1

Substitute $- 1$ into the polynomial.

$3 {(- 1)}^{3} + 10 {(- 1)}^{2} - 1 - 6$

Step 1.1.3.2

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

$3 \cdot - 1 + 10 {(- 1)}^{2} - 1 - 6$

Step 1.1.3.3

Multiply $3$ by $- 1$ .

$- 3 + 10 {(- 1)}^{2} - 1 - 6$

Step 1.1.3.4

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

$- 3 + 10 \cdot 1 - 1 - 6$

Step 1.1.3.5

Multiply $10$ by $1$ .

$- 3 + 10 - 1 - 6$

Step 1.1.3.6

Add $- 3$ and $10$ .

$7 - 1 - 6$

Step 1.1.3.7

Subtract $1$ from $7$ .

$6 - 6$

Step 1.1.3.8

Subtract $6$ from $6$ .

$0$

Step 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.

$\frac{3 x^{3} + 10 x^{2} + x - 6}{x + 1}$

Step 1.1.5

Divide $3 x^{3} + 10 x^{2} + x - 6$ by $x + 1$ .

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Step 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 $0$ .

$x$

$1$

$3 x^{3}$

$10 x^{2}$

$x$

$6$

Step 1.1.5.2

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

					$3 x^{2}$
	$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$

Step 1.1.5.3

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			+	$3 x^{3}$	+	$3 x^{2}$

Step 1.1.5.4

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

				$3 x^{2}$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			-	$3 x^{3}$	-	$3 x^{2}$

Step 1.1.5.5

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

				$3 x^{2}$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$7 x^{2}$

Step 1.1.5.6

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

				$3 x^{2}$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$7 x^{2}$	+	$x$

Step 1.1.5.7

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

				$3 x^{2}$	+	$7 x$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$7 x^{2}$	+	$x$

Step 1.1.5.8

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$7 x$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$7 x^{2}$	+	$x$
					+	$7 x^{2}$	+	$7 x$

Step 1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} + 7 x$

				$3 x^{2}$	+	$7 x$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$7 x^{2}$	+	$x$
					-	$7 x^{2}$	-	$7 x$

Step 1.1.5.10

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

				$3 x^{2}$	+	$7 x$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$7 x^{2}$	+	$x$
					-	$7 x^{2}$	-	$7 x$
							-	$6 x$

Step 1.1.5.11

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

				$3 x^{2}$	+	$7 x$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$7 x^{2}$	+	$x$
					-	$7 x^{2}$	-	$7 x$
							-	$6 x$	-	$6$

Step 1.1.5.12

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

				$3 x^{2}$	+	$7 x$	-	$6$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$7 x^{2}$	+	$x$
					-	$7 x^{2}$	-	$7 x$
							-	$6 x$	-	$6$

Step 1.1.5.13

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$7 x$	-	$6$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$7 x^{2}$	+	$x$
					-	$7 x^{2}$	-	$7 x$
							-	$6 x$	-	$6$
							-	$6 x$	-	$6$

Step 1.1.5.14

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

				$3 x^{2}$	+	$7 x$	-	$6$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$7 x^{2}$	+	$x$
					-	$7 x^{2}$	-	$7 x$
							-	$6 x$	-	$6$
							+	$6 x$	+	$6$

Step 1.1.5.15

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

				$3 x^{2}$	+	$7 x$	-	$6$
$x$	+	$1$		$3 x^{3}$	+	$10 x^{2}$	+	$x$	-	$6$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$7 x^{2}$	+	$x$
					-	$7 x^{2}$	-	$7 x$
							-	$6 x$	-	$6$
							+	$6 x$	+	$6$
										$0$

Step 1.1.5.16

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

$3 x^{2} + 7 x - 6$

Step 1.1.6

Write $3 x^{3} + 10 x^{2} + x - 6$ as a set of factors.

$(x + 1) (3 x^{2} + 7 x - 6) = 0$

Step 1.2

Factor by grouping.

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Step 1.2.1

Factor by grouping.

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Step 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 $a \cdot c = 3 \cdot - 6 = - 18$ and whose sum is $b = 7$ .

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Step 1.2.1.1.1

Factor $7$ out of $7 x$ .

$(x + 1) (3 x^{2} + 7 (x) - 6) = 0$

Step 1.2.1.1.2

Rewrite $7$ as $- 2$ plus $9$

$(x + 1) (3 x^{2} + (- 2 + 9) x - 6) = 0$

Step 1.2.1.1.3

Apply the distributive property.

$(x + 1) (3 x^{2} - 2 x + 9 x - 6) = 0$

Step 1.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.1.2.1

Group the first two terms and the last two terms.

$(x + 1) ((3 x^{2} - 2 x) + 9 x - 6) = 0$

Step 1.2.1.2.2

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

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

Step 1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

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

Step 1.2.2

Remove unnecessary parentheses.

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

Step 2

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

$x + 1 = 0$

$3 x - 2 = 0$

$x + 3 = 0$

Step 3

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

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Step 3.1

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

$x + 1 = 0$

Step 3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4

Set $3 x - 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Step 4.2

Solve $3 x - 2 = 0$ for $x$ .

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Step 4.2.1

Add $2$ to both sides of the equation.

$3 x = 2$

Step 4.2.2

Divide each term in $3 x = 2$ by $3$ and simplify.

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Step 4.2.2.1

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6

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

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

Step 7