Precalculus Examples

$f (x) = x^{5} - 4 x^{4} - x^{3} + 10 x^{2} - 2 x - 4$

Step 1

Set $x^{5} - 4 x^{4} - x^{3} + 10 x^{2} - 2 x - 4$ equal to $0$ .

$x^{5} - 4 x^{4} - x^{3} + 10 x^{2} - 2 x - 4 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Regroup terms.

$x^{5} - x^{3} - 2 x - 4 x^{4} + 10 x^{2} - 4 = 0$

Step 2.1.2

Factor $x$ out of $x^{5} - x^{3} - 2 x$ .

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Step 2.1.2.1

Factor $x$ out of $x^{5}$ .

$x \cdot x^{4} - x^{3} - 2 x - 4 x^{4} + 10 x^{2} - 4 = 0$

Step 2.1.2.2

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

$x \cdot x^{4} + x (- x^{2}) - 2 x - 4 x^{4} + 10 x^{2} - 4 = 0$

Step 2.1.2.3

Factor $x$ out of $- 2 x$ .

$x \cdot x^{4} + x (- x^{2}) + x \cdot - 2 - 4 x^{4} + 10 x^{2} - 4 = 0$

Step 2.1.2.4

Factor $x$ out of $x \cdot x^{4} + x (- x^{2})$ .

$x (x^{4} - x^{2}) + x \cdot - 2 - 4 x^{4} + 10 x^{2} - 4 = 0$

Step 2.1.2.5

Factor $x$ out of $x (x^{4} - x^{2}) + x \cdot - 2$ .

$x (x^{4} - x^{2} - 2) - 4 x^{4} + 10 x^{2} - 4 = 0$

Step 2.1.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x ({(x^{2})}^{2} - x^{2} - 2) - 4 x^{4} + 10 x^{2} - 4 = 0$

Step 2.1.4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (u^{2} - u - 2) - 4 x^{4} + 10 x^{2} - 4 = 0$

Step 2.1.5

Factor $u^{2} - u - 2$ using the AC method.

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Step 2.1.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 2.1.5.2

Write the factored form using these integers.

$x ((u - 2) (u + 1)) - 4 x^{4} + 10 x^{2} - 4 = 0$

Step 2.1.6

Factor.

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Step 2.1.6.1

Replace all occurrences of $u$ with $x^{2}$ .

$x ((x^{2} - 2) (x^{2} + 1)) - 4 x^{4} + 10 x^{2} - 4 = 0$

Step 2.1.6.2

Remove unnecessary parentheses.

$x (x^{2} - 2) (x^{2} + 1) - 4 x^{4} + 10 x^{2} - 4 = 0$

Step 2.1.7

Factor $2$ out of $- 4 x^{4} + 10 x^{2} - 4$ .

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Step 2.1.7.1

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

$x (x^{2} - 2) (x^{2} + 1) + 2 (- 2 x^{4}) + 10 x^{2} - 4 = 0$

Step 2.1.7.2

Factor $2$ out of $10 x^{2}$ .

$x (x^{2} - 2) (x^{2} + 1) + 2 (- 2 x^{4}) + 2 (5 x^{2}) - 4 = 0$

Step 2.1.7.3

Factor $2$ out of $- 4$ .

$x (x^{2} - 2) (x^{2} + 1) + 2 (- 2 x^{4}) + 2 (5 x^{2}) + 2 (- 2) = 0$

Step 2.1.7.4

Factor $2$ out of $2 (- 2 x^{4}) + 2 (5 x^{2})$ .

$x (x^{2} - 2) (x^{2} + 1) + 2 (- 2 x^{4} + 5 x^{2}) + 2 (- 2) = 0$

Step 2.1.7.5

Factor $2$ out of $2 (- 2 x^{4} + 5 x^{2}) + 2 (- 2)$ .

$x (x^{2} - 2) (x^{2} + 1) + 2 (- 2 x^{4} + 5 x^{2} - 2) = 0$

Step 2.1.8

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x (x^{2} - 2) (x^{2} + 1) + 2 (- 2 {(x^{2})}^{2} + 5 x^{2} - 2) = 0$

Step 2.1.9

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (x^{2} - 2) (x^{2} + 1) + 2 (- 2 u^{2} + 5 u - 2) = 0$

Step 2.1.10

Factor by grouping.

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Step 2.1.10.1

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 $a \cdot c = - 2 \cdot - 2 = 4$ and whose sum is $b = 5$ .

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Step 2.1.10.1.1

Factor $5$ out of $5 u$ .

$x (x^{2} - 2) (x^{2} + 1) + 2 (- 2 u^{2} + 5 (u) - 2) = 0$

Step 2.1.10.1.2

Rewrite $5$ as $1$ plus $4$

$x (x^{2} - 2) (x^{2} + 1) + 2 (- 2 u^{2} + (1 + 4) u - 2) = 0$

Step 2.1.10.1.3

Apply the distributive property.

$x (x^{2} - 2) (x^{2} + 1) + 2 (- 2 u^{2} + 1 u + 4 u - 2) = 0$

Step 2.1.10.1.4

Multiply $u$ by $1$ .

$x (x^{2} - 2) (x^{2} + 1) + 2 (- 2 u^{2} + u + 4 u - 2) = 0$

Step 2.1.10.2

Factor out the greatest common factor from each group.

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Step 2.1.10.2.1

Group the first two terms and the last two terms.

$x (x^{2} - 2) (x^{2} + 1) + 2 ((- 2 u^{2} + u) + 4 u - 2) = 0$

Step 2.1.10.2.2

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

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

Step 2.1.10.3

Factor the polynomial by factoring out the greatest common factor, $- 2 u + 1$ .

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

Step 2.1.11

Factor.

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Step 2.1.11.1

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 2.1.11.2

Remove unnecessary parentheses.

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

Step 2.1.12

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

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Step 2.1.12.1

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

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

Step 2.1.12.2

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

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

Step 2.1.12.3

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

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

Step 2.1.13

Apply the distributive property.

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

Step 2.1.14

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

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Step 2.1.14.1

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

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Step 2.1.14.1.1

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

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

Step 2.1.14.1.2

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

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

Step 2.1.14.2

Add $1$ and $2$ .

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

Step 2.1.15

Multiply $x$ by $1$ .

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

Step 2.1.16

Apply the distributive property.

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

Step 2.1.17

Multiply $- 2$ by $2$ .

$(x^{2} - 2) (x^{3} + x - 4 x^{2} + 2 \cdot 1) = 0$

Step 2.1.18

Multiply $2$ by $1$ .

$(x^{2} - 2) (x^{3} + x - 4 x^{2} + 2) = 0$

Step 2.1.19

Reorder terms.

$(x^{2} - 2) (x^{3} - 4 x^{2} + x + 2) = 0$

Step 2.1.20

Factor.

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Step 2.1.20.1

Factor $x^{3} - 4 x^{2} + x + 2$ using the rational roots test.

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Step 2.1.20.1.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1$

Step 2.1.20.1.2

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

$\pm 1, \pm 2$

Step 2.1.20.1.3

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 2.1.20.1.3.1

Substitute $1$ into the polynomial.

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

Step 2.1.20.1.3.2

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

$1 - 4 \cdot 1^{2} + 1 + 2$

Step 2.1.20.1.3.3

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

$1 - 4 \cdot 1 + 1 + 2$

Step 2.1.20.1.3.4

Multiply $- 4$ by $1$ .

$1 - 4 + 1 + 2$

Step 2.1.20.1.3.5

Subtract $4$ from $1$ .

$- 3 + 1 + 2$

Step 2.1.20.1.3.6

Add $- 3$ and $1$ .

$- 2 + 2$

Step 2.1.20.1.3.7

Add $- 2$ and $2$ .

$0$

Step 2.1.20.1.4

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.

$\frac{x^{3} - 4 x^{2} + x + 2}{x - 1}$

Step 2.1.20.1.5

Divide $x^{3} - 4 x^{2} + x + 2$ by $x - 1$ .

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Step 2.1.20.1.5.1

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

$x$

$1$

$x^{3}$

$4 x^{2}$

$x$

$2$

Step 2.1.20.1.5.2

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

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

Step 2.1.20.1.5.3

Multiply the new quotient term by the divisor.

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

Step 2.1.20.1.5.4

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

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

Step 2.1.20.1.5.5

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

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

Step 2.1.20.1.5.6

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

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

Step 2.1.20.1.5.7

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

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

Step 2.1.20.1.5.8

Multiply the new quotient term by the divisor.

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

Step 2.1.20.1.5.9

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

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

Step 2.1.20.1.5.10

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

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

Step 2.1.20.1.5.11

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

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

Step 2.1.20.1.5.12

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

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

Step 2.1.20.1.5.13

Multiply the new quotient term by the divisor.

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

Step 2.1.20.1.5.14

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

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

Step 2.1.20.1.5.15

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

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

Step 2.1.20.1.5.16

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

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

Step 2.1.20.1.6

Write $x^{3} - 4 x^{2} + x + 2$ as a set of factors.

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

Step 2.1.20.2

Remove unnecessary parentheses.

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

Step 2.2

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

$x^{2} - 2 = 0$

$x - 1 = 0$

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

Step 2.3

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

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Step 2.3.1

Set $x^{2} - 2$ equal to $0$ .

$x^{2} - 2 = 0$

Step 2.3.2

Solve $x^{2} - 2 = 0$ for $x$ .

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Step 2.3.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 2.3.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{2}$

Step 2.3.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Step 2.3.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

Step 2.3.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2}, - \sqrt{2}$

Step 2.4

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

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Step 2.4.1

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

$x - 1 = 0$

Step 2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 2.5.2.2

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

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

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

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

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

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.5.2.3.1.2.2

Multiply $- 4$ by $- 2$ .

$x = \frac{3 \pm \sqrt{9 + 8}}{2 \cdot 1}$

Step 2.5.2.3.1.3

Add $9$ and $8$ .

$x = \frac{3 \pm \sqrt{17}}{2 \cdot 1}$

Step 2.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{17}}{2}$

Step 2.5.2.4

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{17}}{2}, \frac{3 - \sqrt{17}}{2}$

Step 2.6

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

$x = \sqrt{2}, - \sqrt{2}, 1, \frac{3 + \sqrt{17}}{2}, \frac{3 - \sqrt{17}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{2}, - \sqrt{2}, 1, \frac{3 + \sqrt{17}}{2}, \frac{3 - \sqrt{17}}{2}$

Decimal Form:

$x = 1.41421356 \dots, - 1.41421356 \dots, 1, 3.56155281 \dots, - 0.56155281 \dots$

Step 4