Algebra Examples

$x^{4} - 4 x^{3} - x^{2} + 16 x - 12$

Step 1

Regroup terms.

$x^{4} - x^{2} - 4 x^{3} + 16 x - 12$

Step 2

Factor $x^{2}$ out of $x^{4} - x^{2}$ .

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Step 2.1

Factor $x^{2}$ out of $x^{4}$ .

$x^{2} x^{2} - x^{2} - 4 x^{3} + 16 x - 12$

Step 2.2

Factor $x^{2}$ out of $- x^{2}$ .

$x^{2} x^{2} + x^{2} \cdot - 1 - 4 x^{3} + 16 x - 12$

Step 2.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 1$ .

$x^{2} (x^{2} - 1) - 4 x^{3} + 16 x - 12$

Step 3

Rewrite $1$ as $1^{2}$ .

$x^{2} (x^{2} - 1^{2}) - 4 x^{3} + 16 x - 12$

Step 4

Factor.

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Step 4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$x^{2} ((x + 1) (x - 1)) - 4 x^{3} + 16 x - 12$

Step 4.2

Remove unnecessary parentheses.

$x^{2} (x + 1) (x - 1) - 4 x^{3} + 16 x - 12$

Step 5

Factor $4$ out of $- 4 x^{3} + 16 x - 12$ .

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Step 5.1

Factor $4$ out of $- 4 x^{3}$ .

$x^{2} (x + 1) (x - 1) + 4 (- x^{3}) + 16 x - 12$

Step 5.2

Factor $4$ out of $16 x$ .

$x^{2} (x + 1) (x - 1) + 4 (- x^{3}) + 4 (4 x) - 12$

Step 5.3

Factor $4$ out of $- 12$ .

$x^{2} (x + 1) (x - 1) + 4 (- x^{3}) + 4 (4 x) + 4 (- 3)$

Step 5.4

Factor $4$ out of $4 (- x^{3}) + 4 (4 x)$ .

$x^{2} (x + 1) (x - 1) + 4 (- x^{3} + 4 x) + 4 (- 3)$

Step 5.5

Factor $4$ out of $4 (- x^{3} + 4 x) + 4 (- 3)$ .

$x^{2} (x + 1) (x - 1) + 4 (- x^{3} + 4 x - 3)$

Step 6

Factor.

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Step 6.1

Factor $- x^{3} + 4 x - 3$ using the rational roots test.

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Step 6.1.1

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 6.1.2

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

$\pm 1, \pm 3$

Step 6.1.3

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 6.1.3.1

Substitute $1$ into the polynomial.

$- 1^{3} + 4 \cdot 1 - 3$

Step 6.1.3.2

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

$- 1 \cdot 1 + 4 \cdot 1 - 3$

Step 6.1.3.3

Multiply $- 1$ by $1$ .

$- 1 + 4 \cdot 1 - 3$

Step 6.1.3.4

Multiply $4$ by $1$ .

$- 1 + 4 - 3$

Step 6.1.3.5

Add $- 1$ and $4$ .

$3 - 3$

Step 6.1.3.6

Subtract $3$ from $3$ .

$0$

Step 6.1.4

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.

$\frac{- x^{3} + 4 x - 3}{x - 1}$

Step 6.1.5

Divide $- x^{3} + 4 x - 3$ by $x - 1$ .

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Step 6.1.5.1

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

$x$

$1$

$x^{3}$

$0 x^{2}$

$4 x$

$3$

Step 6.1.5.2

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

				-	$x^{2}$
	$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$

Step 6.1.5.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			-	$x^{3}$	+	$x^{2}$

Step 6.1.5.4

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

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$

Step 6.1.5.5

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

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$

Step 6.1.5.6

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

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$4 x$

Step 6.1.5.7

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

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$4 x$

Step 6.1.5.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$4 x$
					-	$x^{2}$	+	$x$

Step 6.1.5.9

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

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$x$

Step 6.1.5.10

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

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$x$
							+	$3 x$

Step 6.1.5.11

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

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$x$
							+	$3 x$	-	$3$

Step 6.1.5.12

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

			-	$x^{2}$	-	$x$	+	$3$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$x$
							+	$3 x$	-	$3$

Step 6.1.5.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$x$	+	$3$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$x$
							+	$3 x$	-	$3$
							+	$3 x$	-	$3$

Step 6.1.5.14

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

			-	$x^{2}$	-	$x$	+	$3$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$x$
							+	$3 x$	-	$3$
							-	$3 x$	+	$3$

Step 6.1.5.15

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

			-	$x^{2}$	-	$x$	+	$3$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$4 x$	-	$3$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$4 x$
					+	$x^{2}$	-	$x$
							+	$3 x$	-	$3$
							-	$3 x$	+	$3$
										$0$

Step 6.1.5.16

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

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

Step 6.1.6

Write $- x^{3} + 4 x - 3$ as a set of factors.

$x^{2} (x + 1) (x - 1) + 4 ((x - 1) (- x^{2} - x + 3))$

Step 6.2

Remove unnecessary parentheses.

$x^{2} (x + 1) (x - 1) + 4 (x - 1) (- x^{2} - x + 3)$

Step 7

Factor $x - 1$ out of $x^{2} (x + 1) (x - 1) + 4 (x - 1) (- x^{2} - x + 3)$ .

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Step 7.1

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

$(x - 1) (x^{2} (x + 1)) + 4 (x - 1) (- x^{2} - x + 3)$

Step 7.2

Factor $x - 1$ out of $4 (x - 1) (- x^{2} - x + 3)$ .

$(x - 1) (x^{2} (x + 1)) + (x - 1) (4 (- x^{2} - x + 3))$

Step 7.3

Factor $x - 1$ out of $(x - 1) (x^{2} (x + 1)) + (x - 1) (4 (- x^{2} - x + 3))$ .

$(x - 1) (x^{2} (x + 1) + 4 (- x^{2} - x + 3))$

Step 8

Apply the distributive property.

$(x - 1) (x^{2} x + x^{2} \cdot 1 + 4 (- x^{2} - x + 3))$

Step 9

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

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Step 9.1

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

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Step 9.1.1

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

$(x - 1) (x^{2} x^{1} + x^{2} \cdot 1 + 4 (- x^{2} - x + 3))$

Step 9.1.2

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

$(x - 1) (x^{2 + 1} + x^{2} \cdot 1 + 4 (- x^{2} - x + 3))$

Step 9.2

Add $2$ and $1$ .

$(x - 1) (x^{3} + x^{2} \cdot 1 + 4 (- x^{2} - x + 3))$

Step 10

Multiply $x^{2}$ by $1$ .

$(x - 1) (x^{3} + x^{2} + 4 (- x^{2} - x + 3))$

Step 11

Apply the distributive property.

$(x - 1) (x^{3} + x^{2} + 4 (- x^{2}) + 4 (- x) + 4 \cdot 3)$

Step 12

Simplify.

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Step 12.1

Multiply $- 1$ by $4$ .

$(x - 1) (x^{3} + x^{2} - 4 x^{2} + 4 (- x) + 4 \cdot 3)$

Step 12.2

Multiply $- 1$ by $4$ .

$(x - 1) (x^{3} + x^{2} - 4 x^{2} - 4 x + 4 \cdot 3)$

Step 12.3

Multiply $4$ by $3$ .

$(x - 1) (x^{3} + x^{2} - 4 x^{2} - 4 x + 12)$

Step 13

Subtract $4 x^{2}$ from $x^{2}$ .

$(x - 1) (x^{3} - 3 x^{2} - 4 x + 12)$

Step 14

Factor.

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Step 14.1

Rewrite $x^{3} - 3 x^{2} - 4 x + 12$ in a factored form.

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Step 14.1.1

Factor out the greatest common factor from each group.

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Step 14.1.1.1

Group the first two terms and the last two terms.

$(x - 1) ((x^{3} - 3 x^{2}) - 4 x + 12)$

Step 14.1.1.2

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

$(x - 1) (x^{2} (x - 3) - 4 (x - 3))$

Step 14.1.2

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

$(x - 1) ((x - 3) (x^{2} - 4))$

Step 14.1.3

Rewrite $4$ as $2^{2}$ .

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

Step 14.1.4

Factor.

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Step 14.1.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 14.1.4.2

Remove unnecessary parentheses.

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

Step 14.2

Remove unnecessary parentheses.

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