Precalculus Examples

$x^{5} - 3 x^{4} + 19 x^{3} - 49 x^{2} + 48 x - 16$

Step 1

Set $x^{5} - 3 x^{4} + 19 x^{3} - 49 x^{2} + 48 x - 16$ equal to $0$ .

$x^{5} - 3 x^{4} + 19 x^{3} - 49 x^{2} + 48 x - 16 = 0$

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.

$x^{5} + 19 x^{3} + 48 x - 3 x^{4} - 49 x^{2} - 16 = 0$

Step 2.1.2

Factor $x$ out of $x^{5} + 19 x^{3} + 48 x$ .

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Step 2.1.2.1

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

$x \cdot x^{4} + 19 x^{3} + 48 x - 3 x^{4} - 49 x^{2} - 16 = 0$

Step 2.1.2.2

Factor $x$ out of $19 x^{3}$ .

$x \cdot x^{4} + x (19 x^{2}) + 48 x - 3 x^{4} - 49 x^{2} - 16 = 0$

Step 2.1.2.3

Factor $x$ out of $48 x$ .

$x \cdot x^{4} + x (19 x^{2}) + x \cdot 48 - 3 x^{4} - 49 x^{2} - 16 = 0$

Step 2.1.2.4

Factor $x$ out of $x \cdot x^{4} + x (19 x^{2})$ .

$x (x^{4} + 19 x^{2}) + x \cdot 48 - 3 x^{4} - 49 x^{2} - 16 = 0$

Step 2.1.2.5

Factor $x$ out of $x (x^{4} + 19 x^{2}) + x \cdot 48$ .

$x (x^{4} + 19 x^{2} + 48) - 3 x^{4} - 49 x^{2} - 16 = 0$

Step 2.1.3

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

$x ({(x^{2})}^{2} + 19 x^{2} + 48) - 3 x^{4} - 49 x^{2} - 16 = 0$

Step 2.1.4

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

$x (u^{2} + 19 u + 48) - 3 x^{4} - 49 x^{2} - 16 = 0$

Step 2.1.5

Factor $u^{2} + 19 u + 48$ 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 $48$ and whose sum is $19$ .

$3, 16$

Step 2.1.5.2

Write the factored form using these integers.

$x ((u + 3) (u + 16)) - 3 x^{4} - 49 x^{2} - 16 = 0$

Step 2.1.6

Factor.

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Step 2.1.6.1

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

$x ((x^{2} + 3) (x^{2} + 16)) - 3 x^{4} - 49 x^{2} - 16 = 0$

Step 2.1.6.2

Remove unnecessary parentheses.

$x (x^{2} + 3) (x^{2} + 16) - 3 x^{4} - 49 x^{2} - 16 = 0$

Step 2.1.7

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

$x (x^{2} + 3) (x^{2} + 16) - 3 {(x^{2})}^{2} - 49 x^{2} - 16 = 0$

Step 2.1.8

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

$x (x^{2} + 3) (x^{2} + 16) - 3 u^{2} - 49 u - 16 = 0$

Step 2.1.9

Factor by grouping.

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Step 2.1.9.1

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 $a \cdot c = - 3 \cdot - 16 = 48$ and whose sum is $b = - 49$ .

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Step 2.1.9.1.1

Factor $- 49$ out of $- 49 u$ .

$x (x^{2} + 3) (x^{2} + 16) - 3 u^{2} - 49 u - 16 = 0$

Step 2.1.9.1.2

Rewrite $- 49$ as $- 1$ plus $- 48$

$x (x^{2} + 3) (x^{2} + 16) - 3 u^{2} + (- 1 - 48) u - 16 = 0$

Step 2.1.9.1.3

Apply the distributive property.

$x (x^{2} + 3) (x^{2} + 16) - 3 u^{2} - 1 u - 48 u - 16 = 0$

Step 2.1.9.2

Factor out the greatest common factor from each group.

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Step 2.1.9.2.1

Group the first two terms and the last two terms.

$x (x^{2} + 3) (x^{2} + 16) - 3 u^{2} - 1 u - 48 u - 16 = 0$

Step 2.1.9.2.2

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

$x (x^{2} + 3) (x^{2} + 16) + u (- 3 u - 1) + 16 (- 3 u - 1) = 0$

Step 2.1.9.3

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

$x (x^{2} + 3) (x^{2} + 16) + (- 3 u - 1) (u + 16) = 0$

Step 2.1.10

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

$x (x^{2} + 3) (x^{2} + 16) + (- 3 x^{2} - 1) (x^{2} + 16) = 0$

Step 2.1.11

Factor $x^{2} + 16$ out of $x (x^{2} + 3) (x^{2} + 16) + (- 3 x^{2} - 1) (x^{2} + 16)$ .

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Step 2.1.11.1

Factor $x^{2} + 16$ out of $x (x^{2} + 3) (x^{2} + 16)$ .

$(x^{2} + 16) (x (x^{2} + 3)) + (- 3 x^{2} - 1) (x^{2} + 16) = 0$

Step 2.1.11.2

Factor $x^{2} + 16$ out of $(- 3 x^{2} - 1) (x^{2} + 16)$ .

$(x^{2} + 16) (x (x^{2} + 3)) + (x^{2} + 16) (- 3 x^{2} - 1) = 0$

Step 2.1.11.3

Factor $x^{2} + 16$ out of $(x^{2} + 16) (x (x^{2} + 3)) + (x^{2} + 16) (- 3 x^{2} - 1)$ .

$(x^{2} + 16) (x (x^{2} + 3) - 3 x^{2} - 1) = 0$

Step 2.1.12

Apply the distributive property.

$(x^{2} + 16) (x \cdot x^{2} + x \cdot 3 - 3 x^{2} - 1) = 0$

Step 2.1.13

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

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Step 2.1.13.1

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

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Step 2.1.13.1.1

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

$(x^{2} + 16) (x \cdot x^{2} + x \cdot 3 - 3 x^{2} - 1) = 0$

Step 2.1.13.1.2

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

$(x^{2} + 16) (x^{1 + 2} + x \cdot 3 - 3 x^{2} - 1) = 0$

Step 2.1.13.2

Add $1$ and $2$ .

$(x^{2} + 16) (x^{3} + x \cdot 3 - 3 x^{2} - 1) = 0$

Step 2.1.14

Move $3$ to the left of $x$ .

$(x^{2} + 16) (x^{3} + 3 x - 3 x^{2} - 1) = 0$

Step 2.1.15

Reorder terms.

$(x^{2} + 16) (x^{3} - 3 x^{2} + 3 x - 1) = 0$

Step 2.1.16

Rewrite $x^{3} - 3 x^{2} + 3 x - 1$ in a factored form.

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Step 2.1.16.1

Factor $x^{3} - 3 x^{2} + 3 x - 1$ using the rational roots test.

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Step 2.1.16.1.1

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.

$p = \pm 1$

$q = \pm 1$

Step 2.1.16.1.2

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

$\pm 1$

Step 2.1.16.1.3

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 2.1.16.1.3.1

Substitute $1$ into the polynomial.

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

Step 2.1.16.1.3.2

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

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

Step 2.1.16.1.3.3

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

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

Step 2.1.16.1.3.4

Multiply $- 3$ by $1$ .

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

Step 2.1.16.1.3.5

Subtract $3$ from $1$ .

$- 2 + 3 \cdot 1 - 1$

Step 2.1.16.1.3.6

Multiply $3$ by $1$ .

$- 2 + 3 - 1$

Step 2.1.16.1.3.7

Add $- 2$ and $3$ .

$1 - 1$

Step 2.1.16.1.3.8

Subtract $1$ from $1$ .

$0$

Step 2.1.16.1.4

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.

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

Step 2.1.16.1.5

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

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Step 2.1.16.1.5.1

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

$x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

Step 2.1.16.1.5.2

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

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

Step 2.1.16.1.5.3

Multiply the new quotient term by the divisor.

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

Step 2.1.16.1.5.4

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

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

Step 2.1.16.1.5.5

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

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

Step 2.1.16.1.5.6

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

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

Step 2.1.16.1.5.7

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

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

Step 2.1.16.1.5.8

Multiply the new quotient term by the divisor.

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

Step 2.1.16.1.5.9

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

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

Step 2.1.16.1.5.10

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

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

Step 2.1.16.1.5.11

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

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

Step 2.1.16.1.5.12

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

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

Step 2.1.16.1.5.13

Multiply the new quotient term by the divisor.

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

Step 2.1.16.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $x - 1$

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

Step 2.1.16.1.5.15

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

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

Step 2.1.16.1.5.16

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

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

Step 2.1.16.1.6

Write $x^{3} - 3 x^{2} + 3 x - 1$ as a set of factors.

$(x^{2} + 16) ((x - 1) (x^{2} - 2 x + 1)) = 0$

Step 2.1.16.2

Factor using the perfect square rule.

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Step 2.1.16.2.1

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

$(x^{2} + 16) ((x - 1) (x^{2} - 2 x + 1^{2})) = 0$

Step 2.1.16.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 2.1.16.2.3

Rewrite the polynomial.

$(x^{2} + 16) ((x - 1) (x^{2} - 2 \cdot x \cdot 1 + 1^{2})) = 0$

Step 2.1.16.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$(x^{2} + 16) ((x - 1) {(x - 1)}^{2}) = 0$

Step 2.1.16.3

Combine like factors.

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Step 2.1.16.3.1

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

$(x^{2} + 16) ((x - 1) {(x - 1)}^{2}) = 0$

Step 2.1.16.3.2

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

$(x^{2} + 16) {(x - 1)}^{1 + 2} = 0$

Step 2.1.16.3.3

Add $1$ and $2$ .

$(x^{2} + 16) {(x - 1)}^{3} = 0$

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 $0$ .

$x^{2} + 16 = 0$

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

Step 2.3

Set $x^{2} + 16$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{2} + 16$ equal to $0$ .

$x^{2} + 16 = 0$

Step 2.3.2

Solve $x^{2} + 16 = 0$ for $x$ .

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Step 2.3.2.1

Subtract $16$ from both sides of the equation.

$x^{2} = - 16$

Step 2.3.2.2

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

$x = \pm \sqrt{- 16}$

Step 2.3.2.3

Simplify $\pm \sqrt{- 16}$ .

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Step 2.3.2.3.1

Rewrite $- 16$ as $- 1 (16)$ .

$x = \pm \sqrt{- 1 (16)}$

Step 2.3.2.3.2

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{16}$

Step 2.3.2.3.3

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

$x = \pm i \cdot \sqrt{16}$

Step 2.3.2.3.4

Rewrite $16$ as $4^{2}$ .

$x = \pm i \cdot \sqrt{4^{2}}$

Step 2.3.2.3.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 4$

Step 2.3.2.3.6

Move $4$ to the left of $i$ .

$x = \pm 4 i$

Step 2.3.2.4

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

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Step 2.3.2.4.1

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

$x = 4 i$

Step 2.3.2.4.2

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

$x = - 4 i$

Step 2.3.2.4.3

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

$x = 4 i, - 4 i$

Step 2.4

Set ${(x - 1)}^{3}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set ${(x - 1)}^{3}$ equal to $0$ .

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

Step 2.4.2

Solve ${(x - 1)}^{3} = 0$ for $x$ .

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Step 2.4.2.1

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

$x - 1 = 0$

Step 2.4.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.5

The final solution is all the values that make $(x^{2} + 16) {(x - 1)}^{3} = 0$ true.

$x = 4 i, - 4 i, 1$

Step 3