Pre-Algebra Examples

$2 x^{3} + 5 x^{2} + 3 x - 3 = 0$

Step 1

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

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Step 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.

$p = \pm 1, \pm 3$

$q = \pm 1, \pm 2$

Step 1.2

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

$\pm 1, \pm 0.5, \pm 3, \pm 1.5$

Step 1.3

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

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Step 1.3.1

Substitute $0.5$ into the polynomial.

$2 \cdot {0.5}^{3} + 5 \cdot {0.5}^{2} + 3 \cdot 0.5 - 3$

Step 1.3.2

Raise $0.5$ to the power of $3$ .

$2 \cdot 0.125 + 5 \cdot {0.5}^{2} + 3 \cdot 0.5 - 3$

Step 1.3.3

Multiply $2$ by $0.125$ .

$0.25 + 5 \cdot {0.5}^{2} + 3 \cdot 0.5 - 3$

Step 1.3.4

Raise $0.5$ to the power of $2$ .

$0.25 + 5 \cdot 0.25 + 3 \cdot 0.5 - 3$

Step 1.3.5

Multiply $5$ by $0.25$ .

$0.25 + 1.25 + 3 \cdot 0.5 - 3$

Step 1.3.6

Add $0.25$ and $1.25$ .

$1.5 + 3 \cdot 0.5 - 3$

Step 1.3.7

Multiply $3$ by $0.5$ .

$1.5 + 1.5 - 3$

Step 1.3.8

Add $1.5$ and $1.5$ .

$3 - 3$

Step 1.3.9

Subtract $3$ from $3$ .

$0$

Step 1.4

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} + 5 x^{2} + 3 x - 3}{2 x - 1}$

Step 1.5

Divide $2 x^{3} + 5 x^{2} + 3 x - 3$ by $2 x - 1$ .

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Step 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 $0$ .

$2 x$

$1$

$2 x^{3}$

$5 x^{2}$

$3 x$

$3$

Step 1.5.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $2 x$ .

					$x^{2}$
	$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	+	$3 x$	-	$3$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	+	$3 x$	-	$3$
			+	$2 x^{3}$	-	$x^{2}$

Step 1.5.4

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

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	+	$3 x$	-	$3$
			-	$2 x^{3}$	+	$x^{2}$

Step 1.5.5

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

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

Step 1.5.6

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

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

Step 1.5.7

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $2 x$ .

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

Step 1.5.8

Multiply the new quotient term by the divisor.

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

Step 1.5.9

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

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

Step 1.5.10

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

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

Step 1.5.11

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

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

Step 1.5.12

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $2 x$ .

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

Step 1.5.13

Multiply the new quotient term by the divisor.

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

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $6 x - 3$

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

Step 1.5.15

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

				$x^{2}$	+	$3 x$	+	$3$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	+	$3 x$	-	$3$
			-	$2 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	+	$3 x$
					-	$6 x^{2}$	+	$3 x$
							+	$6 x$	-	$3$
							-	$6 x$	+	$3$
										$0$

Step 1.5.16

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

$x^{2} + 3 x + 3$

Step 1.6

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

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

Step 2

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

$2 x - 1 = 0$

$x^{2} + 3 x + 3 = 0$

Step 3

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 3.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 3.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 3.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 4

Set $x^{2} + 3 x + 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} + 3 x + 3$ equal to $0$ .

$x^{2} + 3 x + 3 = 0$

Step 4.2

Solve $x^{2} + 3 x + 3 = 0$ for $x$ .

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Step 4.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Step 4.2.3

Simplify.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 3}}{2 \cdot 1}$

Step 4.2.3.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 3 \pm \sqrt{9 - 12}}{2 \cdot 1}$

Step 4.2.3.1.3

Subtract $12$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 4.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 4.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 4.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \sqrt{3}}{2 \cdot 1}$

Step 4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{3}}{2}$

Step 4.2.4

The final answer is the combination of both solutions.

$x = - \frac{3 - i \sqrt{3}}{2}, - \frac{3 + i \sqrt{3}}{2}$

Step 5

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

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