Algebra Examples

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

Step 1

Factor $2 x$ out of $6 x {(3 x - 1)}^{2} + 2 x^{2} {(1 - 3 x)}^{2}$ .

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Step 1.1

Factor $2 x$ out of $6 x {(3 x - 1)}^{2}$ .

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

Step 1.2

Factor $2 x$ out of $2 x^{2} {(1 - 3 x)}^{2}$ .

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

Step 1.3

Factor $2 x$ out of $2 x (3 {(3 x - 1)}^{2}) + 2 x (x {(1 - 3 x)}^{2})$ .

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

Step 2

Rewrite ${(3 x - 1)}^{2}$ as $(3 x - 1) (3 x - 1)$ .

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

Step 3

Expand $(3 x - 1) (3 x - 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.2.1

Move $x$ .

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

Step 4.1.2.2

Multiply $x$ by $x$ .

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

Step 4.1.3

Multiply $3$ by $3$ .

$2 x (3 (9 x^{2} + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1) + x {(1 - 3 x)}^{2})$

Step 4.1.4

Multiply $- 1$ by $3$ .

$2 x (3 (9 x^{2} - 3 x - 1 (3 x) - 1 \cdot - 1) + x {(1 - 3 x)}^{2})$

Step 4.1.5

Multiply $3$ by $- 1$ .

$2 x (3 (9 x^{2} - 3 x - 3 x - 1 \cdot - 1) + x {(1 - 3 x)}^{2})$

Step 4.1.6

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Subtract $3 x$ from $- 3 x$ .

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

Step 5

Apply the distributive property.

$2 x (3 (9 x^{2}) + 3 (- 6 x) + 3 \cdot 1 + x {(1 - 3 x)}^{2})$

Step 6

Simplify.

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Step 6.1

Multiply $9$ by $3$ .

$2 x (27 x^{2} + 3 (- 6 x) + 3 \cdot 1 + x {(1 - 3 x)}^{2})$

Step 6.2

Multiply $- 6$ by $3$ .

$2 x (27 x^{2} - 18 x + 3 \cdot 1 + x {(1 - 3 x)}^{2})$

Step 6.3

Multiply $3$ by $1$ .

$2 x (27 x^{2} - 18 x + 3 + x {(1 - 3 x)}^{2})$

Step 7

Rewrite ${(1 - 3 x)}^{2}$ as $(1 - 3 x) (1 - 3 x)$ .

$2 x (27 x^{2} - 18 x + 3 + x ((1 - 3 x) (1 - 3 x)))$

Step 8

Expand $(1 - 3 x) (1 - 3 x)$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$2 x (27 x^{2} - 18 x + 3 + x (1 (1 - 3 x) - 3 x (1 - 3 x)))$

Step 8.2

Apply the distributive property.

$2 x (27 x^{2} - 18 x + 3 + x (1 \cdot 1 + 1 (- 3 x) - 3 x (1 - 3 x)))$

Step 8.3

Apply the distributive property.

$2 x (27 x^{2} - 18 x + 3 + x (1 \cdot 1 + 1 (- 3 x) - 3 x \cdot 1 - 3 x (- 3 x)))$

Step 9

Simplify and combine like terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $1$ by $1$ .

$2 x (27 x^{2} - 18 x + 3 + x (1 + 1 (- 3 x) - 3 x \cdot 1 - 3 x (- 3 x)))$

Step 9.1.2

Multiply $- 3 x$ by $1$ .

$2 x (27 x^{2} - 18 x + 3 + x (1 - 3 x - 3 x \cdot 1 - 3 x (- 3 x)))$

Step 9.1.3

Multiply $- 3$ by $1$ .

$2 x (27 x^{2} - 18 x + 3 + x (1 - 3 x - 3 x - 3 x (- 3 x)))$

Step 9.1.4

Rewrite using the commutative property of multiplication.

$2 x (27 x^{2} - 18 x + 3 + x (1 - 3 x - 3 x - 3 \cdot - 3 x \cdot x))$

Step 9.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 9.1.5.1

Move $x$ .

$2 x (27 x^{2} - 18 x + 3 + x (1 - 3 x - 3 x - 3 \cdot - 3 (x \cdot x)))$

Step 9.1.5.2

Multiply $x$ by $x$ .

$2 x (27 x^{2} - 18 x + 3 + x (1 - 3 x - 3 x - 3 \cdot - 3 x^{2}))$

Step 9.1.6

Multiply $- 3$ by $- 3$ .

$2 x (27 x^{2} - 18 x + 3 + x (1 - 3 x - 3 x + 9 x^{2}))$

Step 9.2

Subtract $3 x$ from $- 3 x$ .

$2 x (27 x^{2} - 18 x + 3 + x (1 - 6 x + 9 x^{2}))$

Step 10

Apply the distributive property.

$2 x (27 x^{2} - 18 x + 3 + x \cdot 1 + x (- 6 x) + x (9 x^{2}))$

Step 11

Simplify.

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Step 11.1

Multiply $x$ by $1$ .

$2 x (27 x^{2} - 18 x + 3 + x + x (- 6 x) + x (9 x^{2}))$

Step 11.2

Rewrite using the commutative property of multiplication.

$2 x (27 x^{2} - 18 x + 3 + x - 6 x \cdot x + x (9 x^{2}))$

Step 11.3

Rewrite using the commutative property of multiplication.

$2 x (27 x^{2} - 18 x + 3 + x - 6 x \cdot x + 9 x \cdot x^{2})$

Step 12

Simplify each term.

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Step 12.1

Multiply $x$ by $x$ by adding the exponents.

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Step 12.1.1

Move $x$ .

$2 x (27 x^{2} - 18 x + 3 + x - 6 (x \cdot x) + 9 x \cdot x^{2})$

Step 12.1.2

Multiply $x$ by $x$ .

$2 x (27 x^{2} - 18 x + 3 + x - 6 x^{2} + 9 x \cdot x^{2})$

Step 12.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 12.2.1

Move $x^{2}$ .

$2 x (27 x^{2} - 18 x + 3 + x - 6 x^{2} + 9 (x^{2} x))$

Step 12.2.2

Multiply $x^{2}$ by $x$ .

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Step 12.2.2.1

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

$2 x (27 x^{2} - 18 x + 3 + x - 6 x^{2} + 9 (x^{2} x^{1}))$

Step 12.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x (27 x^{2} - 18 x + 3 + x - 6 x^{2} + 9 x^{2 + 1})$

Step 12.2.3

Add $2$ and $1$ .

$2 x (27 x^{2} - 18 x + 3 + x - 6 x^{2} + 9 x^{3})$

Step 13

Subtract $6 x^{2}$ from $27 x^{2}$ .

$2 x (21 x^{2} - 18 x + 3 + x + 9 x^{3})$

Step 14

Add $- 18 x$ and $x$ .

$2 x (21 x^{2} - 17 x + 3 + 9 x^{3})$

Step 15

Reorder terms.

$2 x (9 x^{3} + 21 x^{2} - 17 x + 3)$

Step 16

Factor.

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Step 16.1

Rewrite $9 x^{3} + 21 x^{2} - 17 x + 3$ in a factored form.

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Step 16.1.1

Factor $9 x^{3} + 21 x^{2} - 17 x + 3$ using the rational roots test.

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Step 16.1.1.1

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.

$p = \pm 1, \pm 3$

$q = \pm 1, \pm 9, \pm 3$

Step 16.1.1.2

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

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

Step 16.1.1.3

Substitute $0.\overline{3}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.\overline{3}$ is a root of the polynomial.

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Step 16.1.1.3.1

Substitute $0.\overline{3}$ into the polynomial.

$9 \cdot {0.\overline{3}}^{3} + 21 \cdot {0.\overline{3}}^{2} - 17 \cdot 0.\overline{3} + 3$

Step 16.1.1.3.2

Raise $0.\overline{3}$ to the power of $3$ .

$9 \cdot 0.\overline{037} + 21 \cdot {0.\overline{3}}^{2} - 17 \cdot 0.\overline{3} + 3$

Step 16.1.1.3.3

Multiply $9$ by $0.\overline{037}$ .

$0.\overline{3} + 21 \cdot {0.\overline{3}}^{2} - 17 \cdot 0.\overline{3} + 3$

Step 16.1.1.3.4

Raise $0.\overline{3}$ to the power of $2$ .

$0.\overline{3} + 21 \cdot 0.\overline{1} - 17 \cdot 0.\overline{3} + 3$

Step 16.1.1.3.5

Multiply $21$ by $0.\overline{1}$ .

$0.\overline{3} + 2.\overline{3} - 17 \cdot 0.\overline{3} + 3$

Step 16.1.1.3.6

Add $0.\overline{3}$ and $2.\overline{3}$ .

$2.\overline{6} - 17 \cdot 0.\overline{3} + 3$

Step 16.1.1.3.7

Multiply $- 17$ by $0.\overline{3}$ .

$2.\overline{6} - 5.\overline{6} + 3$

Step 16.1.1.3.8

Subtract $5.\overline{6}$ from $2.\overline{6}$ .

$- 3 + 3$

Step 16.1.1.3.9

Add $- 3$ and $3$ .

$0$

Step 16.1.1.4

Since $0.\overline{3}$ is a known root, divide the polynomial by $3 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{9 x^{3} + 21 x^{2} - 17 x + 3}{3 x - 1}$

Step 16.1.1.5

Divide $9 x^{3} + 21 x^{2} - 17 x + 3$ by $3 x - 1$ .

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Step 16.1.1.5.1

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

$3 x$

$1$

$9 x^{3}$

$21 x^{2}$

$17 x$

$3$

Step 16.1.1.5.2

Divide the highest order term in the dividend $9 x^{3}$ by the highest order term in divisor $3 x$ .

					$3 x^{2}$
	$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$

Step 16.1.1.5.3

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			+	$9 x^{3}$	-	$3 x^{2}$

Step 16.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{3} - 3 x^{2}$

				$3 x^{2}$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			-	$9 x^{3}$	+	$3 x^{2}$

Step 16.1.1.5.5

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

				$3 x^{2}$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$24 x^{2}$

Step 16.1.1.5.6

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

				$3 x^{2}$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$24 x^{2}$	-	$17 x$

Step 16.1.1.5.7

Divide the highest order term in the dividend $24 x^{2}$ by the highest order term in divisor $3 x$ .

				$3 x^{2}$	+	$8 x$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$24 x^{2}$	-	$17 x$

Step 16.1.1.5.8

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$8 x$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$24 x^{2}$	-	$17 x$
					+	$24 x^{2}$	-	$8 x$

Step 16.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $24 x^{2} - 8 x$

				$3 x^{2}$	+	$8 x$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$24 x^{2}$	-	$17 x$
					-	$24 x^{2}$	+	$8 x$

Step 16.1.1.5.10

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

				$3 x^{2}$	+	$8 x$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$24 x^{2}$	-	$17 x$
					-	$24 x^{2}$	+	$8 x$
							-	$9 x$

Step 16.1.1.5.11

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

				$3 x^{2}$	+	$8 x$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$24 x^{2}$	-	$17 x$
					-	$24 x^{2}$	+	$8 x$
							-	$9 x$	+	$3$

Step 16.1.1.5.12

Divide the highest order term in the dividend $- 9 x$ by the highest order term in divisor $3 x$ .

				$3 x^{2}$	+	$8 x$	-	$3$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$24 x^{2}$	-	$17 x$
					-	$24 x^{2}$	+	$8 x$
							-	$9 x$	+	$3$

Step 16.1.1.5.13

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$8 x$	-	$3$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$24 x^{2}$	-	$17 x$
					-	$24 x^{2}$	+	$8 x$
							-	$9 x$	+	$3$
							-	$9 x$	+	$3$

Step 16.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x + 3$

				$3 x^{2}$	+	$8 x$	-	$3$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$24 x^{2}$	-	$17 x$
					-	$24 x^{2}$	+	$8 x$
							-	$9 x$	+	$3$
							+	$9 x$	-	$3$

Step 16.1.1.5.15

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

				$3 x^{2}$	+	$8 x$	-	$3$
$3 x$	-	$1$		$9 x^{3}$	+	$21 x^{2}$	-	$17 x$	+	$3$
			-	$9 x^{3}$	+	$3 x^{2}$
					+	$24 x^{2}$	-	$17 x$
					-	$24 x^{2}$	+	$8 x$
							-	$9 x$	+	$3$
							+	$9 x$	-	$3$
										$0$

Step 16.1.1.5.16

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

$3 x^{2} + 8 x - 3$

Step 16.1.1.6

Write $9 x^{3} + 21 x^{2} - 17 x + 3$ as a set of factors.

$2 x ((3 x - 1) (3 x^{2} + 8 x - 3))$

Step 16.1.2

Factor by grouping.

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Step 16.1.2.1

Factor by grouping.

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Step 16.1.2.1.1

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 $a \cdot c = 3 \cdot - 3 = - 9$ and whose sum is $b = 8$ .

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Step 16.1.2.1.1.1

Factor $8$ out of $8 x$ .

$2 x ((3 x - 1) (3 x^{2} + 8 (x) - 3))$

Step 16.1.2.1.1.2

Rewrite $8$ as $- 1$ plus $9$

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

Step 16.1.2.1.1.3

Apply the distributive property.

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

Step 16.1.2.1.2

Factor out the greatest common factor from each group.

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Step 16.1.2.1.2.1

Group the first two terms and the last two terms.

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

Step 16.1.2.1.2.2

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

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

Step 16.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

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

Step 16.1.2.2

Remove unnecessary parentheses.

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

Step 16.1.3

Combine like factors.

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Step 16.1.3.1

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

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

Step 16.1.3.2

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

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

Step 16.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 16.1.3.4

Add $1$ and $1$ .

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

Step 16.2

Remove unnecessary parentheses.

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