Basic Math Examples

$3 a^{4} - 6 a^{3} + 9 a$

Step 1

Factor $3 a$ out of $3 a^{4} - 6 a^{3} + 9 a$ .

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Step 1.1

Factor $3 a$ out of $3 a^{4}$ .

$3 a (a^{3}) - 6 a^{3} + 9 a$

Step 1.2

Factor $3 a$ out of $- 6 a^{3}$ .

$3 a (a^{3}) + 3 a (- 2 a^{2}) + 9 a$

Step 1.3

Factor $3 a$ out of $9 a$ .

$3 a (a^{3}) + 3 a (- 2 a^{2}) + 3 a (3)$

Step 1.4

Factor $3 a$ out of $3 a (a^{3}) + 3 a (- 2 a^{2})$ .

$3 a (a^{3} - 2 a^{2}) + 3 a (3)$

Step 1.5

Factor $3 a$ out of $3 a (a^{3} - 2 a^{2}) + 3 a (3)$ .

$3 a (a^{3} - 2 a^{2} + 3)$

Step 2

Factor.

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Step 2.1

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

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Step 2.1.1

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.

$p = \pm 1, \pm 3$

$q = \pm 1$

Step 2.1.2

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

$\pm 1, \pm 3$

Step 2.1.3

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 2.1.3.1

Substitute $- 1$ into the polynomial.

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

$- 1 - 2 \cdot 1 + 3$

Step 2.1.3.4

Multiply $- 2$ by $1$ .

$- 1 - 2 + 3$

Step 2.1.3.5

Subtract $2$ from $- 1$ .

$- 3 + 3$

Step 2.1.3.6

Add $- 3$ and $3$ .

$0$

Step 2.1.4

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

$\frac{a^{3} - 2 a^{2} + 3}{a + 1}$

Step 2.1.5

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

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Step 2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$a$

$1$

$a^{3}$

$2 a^{2}$

$0 a$

$3$

Step 2.1.5.2

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

					$a^{2}$
	$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

				$a^{2}$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			+	$a^{3}$	+	$a^{2}$

Step 2.1.5.4

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

				$a^{2}$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			-	$a^{3}$	-	$a^{2}$

Step 2.1.5.5

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

				$a^{2}$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			-	$a^{3}$	-	$a^{2}$
					-	$3 a^{2}$

Step 2.1.5.6

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

				$a^{2}$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			-	$a^{3}$	-	$a^{2}$
					-	$3 a^{2}$	+	$0 a$

Step 2.1.5.7

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

				$a^{2}$	-	$3 a$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			-	$a^{3}$	-	$a^{2}$
					-	$3 a^{2}$	+	$0 a$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

				$a^{2}$	-	$3 a$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			-	$a^{3}$	-	$a^{2}$
					-	$3 a^{2}$	+	$0 a$
					-	$3 a^{2}$	-	$3 a$

Step 2.1.5.9

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

				$a^{2}$	-	$3 a$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			-	$a^{3}$	-	$a^{2}$
					-	$3 a^{2}$	+	$0 a$
					+	$3 a^{2}$	+	$3 a$

Step 2.1.5.10

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

				$a^{2}$	-	$3 a$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			-	$a^{3}$	-	$a^{2}$
					-	$3 a^{2}$	+	$0 a$
					+	$3 a^{2}$	+	$3 a$
							+	$3 a$

Step 2.1.5.11

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

				$a^{2}$	-	$3 a$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			-	$a^{3}$	-	$a^{2}$
					-	$3 a^{2}$	+	$0 a$
					+	$3 a^{2}$	+	$3 a$
							+	$3 a$	+	$3$

Step 2.1.5.12

Divide the highest order term in the dividend $3 a$ by the highest order term in divisor $a$ .

				$a^{2}$	-	$3 a$	+	$3$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			-	$a^{3}$	-	$a^{2}$
					-	$3 a^{2}$	+	$0 a$
					+	$3 a^{2}$	+	$3 a$
							+	$3 a$	+	$3$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

				$a^{2}$	-	$3 a$	+	$3$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			-	$a^{3}$	-	$a^{2}$
					-	$3 a^{2}$	+	$0 a$
					+	$3 a^{2}$	+	$3 a$
							+	$3 a$	+	$3$
							+	$3 a$	+	$3$

Step 2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $3 a + 3$

				$a^{2}$	-	$3 a$	+	$3$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			-	$a^{3}$	-	$a^{2}$
					-	$3 a^{2}$	+	$0 a$
					+	$3 a^{2}$	+	$3 a$
							+	$3 a$	+	$3$
							-	$3 a$	-	$3$

Step 2.1.5.15

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

				$a^{2}$	-	$3 a$	+	$3$
$a$	+	$1$		$a^{3}$	-	$2 a^{2}$	+	$0 a$	+	$3$
			-	$a^{3}$	-	$a^{2}$
					-	$3 a^{2}$	+	$0 a$
					+	$3 a^{2}$	+	$3 a$
							+	$3 a$	+	$3$
							-	$3 a$	-	$3$
										$0$

Step 2.1.5.16

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

$a^{2} - 3 a + 3$

Step 2.1.6

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

$3 a ((a + 1) (a^{2} - 3 a + 3))$

Step 2.2

Remove unnecessary parentheses.

$3 a (a + 1) (a^{2} - 3 a + 3)$