Precalculus Examples

$f (x) = x^{4} - 3 x^{3} - 6 x^{2} + 6 x + 8$

Step 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.

$p = \pm 1, \pm 2, \pm 4, \pm 8$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 4, \pm 8$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(- 1)}^{4} - 3 {(- 1)}^{3} - 6 {(- 1)}^{2} + 6 (- 1) + 8$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 1$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise $- 1$ to the power of $4$ .

$1 - 3 {(- 1)}^{3} - 6 {(- 1)}^{2} + 6 (- 1) + 8$

Step 4.1.2

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

$1 - 3 \cdot - 1 - 6 {(- 1)}^{2} + 6 (- 1) + 8$

Step 4.1.3

Multiply $- 3$ by $- 1$ .

$1 + 3 - 6 {(- 1)}^{2} + 6 (- 1) + 8$

Step 4.1.4

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

$1 + 3 - 6 \cdot 1 + 6 (- 1) + 8$

Step 4.1.5

Multiply $- 6$ by $1$ .

$1 + 3 - 6 + 6 (- 1) + 8$

Step 4.1.6

Multiply $6$ by $- 1$ .

$1 + 3 - 6 - 6 + 8$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Add $1$ and $3$ .

$4 - 6 - 6 + 8$

Step 4.2.2

Subtract $6$ from $4$ .

$- 2 - 6 + 8$

Step 4.2.3

Subtract $6$ from $- 2$ .

$- 8 + 8$

Step 4.2.4

Add $- 8$ and $8$ .

$0$

Step 5

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.

$\frac{x^{4} - 3 x^{3} - 6 x^{2} + 6 x + 8}{x + 1}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Step 6.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 1$	$1$	$- 3$	$- 6$	$6$	$8$

Step 6.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$1$	$- 3$	$- 6$	$6$	$8$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(- 1)$ and place the result of $(- 1)$ under the next term in the dividend $(- 3)$ .

$- 1$	$1$	$- 3$	$- 6$	$6$	$8$
		$- 1$
	$1$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$- 3$	$- 6$	$6$	$8$
		$- 1$
	$1$	$- 4$

Step 6.5

Multiply the newest entry in the result $(- 4)$ by the divisor $(- 1)$ and place the result of $(4)$ under the next term in the dividend $(- 6)$ .

$- 1$	$1$	$- 3$	$- 6$	$6$	$8$
		$- 1$	$4$
	$1$	$- 4$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$- 3$	$- 6$	$6$	$8$
		$- 1$	$4$
	$1$	$- 4$	$- 2$

Step 6.7

Multiply the newest entry in the result $(- 2)$ by the divisor $(- 1)$ and place the result of $(2)$ under the next term in the dividend $(6)$ .

$- 1$	$1$	$- 3$	$- 6$	$6$	$8$
		$- 1$	$4$	$2$
	$1$	$- 4$	$- 2$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$- 3$	$- 6$	$6$	$8$
		$- 1$	$4$	$2$
	$1$	$- 4$	$- 2$	$8$

Step 6.9

Multiply the newest entry in the result $(8)$ by the divisor $(- 1)$ and place the result of $(- 8)$ under the next term in the dividend $(8)$ .

$- 1$	$1$	$- 3$	$- 6$	$6$	$8$
		$- 1$	$4$	$2$	$- 8$
	$1$	$- 4$	$- 2$	$8$

Step 6.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$- 3$	$- 6$	$6$	$8$
		$- 1$	$4$	$2$	$- 8$
	$1$	$- 4$	$- 2$	$8$	$0$

Step 6.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 4 x^{2} + (- 2) x + 8$

Step 6.12

Simplify the quotient polynomial.

$x^{3} - 4 x^{2} - 2 x + 8$

Step 7

Factor out the greatest common factor from each group.

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Step 7.1

Group the first two terms and the last two terms.

$(x + 1) (x^{3} - 4 x^{2}) - 2 x + 8$

Step 7.2

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

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

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

Step 8

Factor the polynomial by factoring out the greatest common factor, $x - 4$ .

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

Step 9

Factor the left side of the equation.

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Step 9.1

Regroup terms.

$- 3 x^{3} + 6 x + x^{4} - 6 x^{2} + 8 = 0$

Step 9.2

Factor $- 3 x$ out of $- 3 x^{3} + 6 x$ .

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Step 9.2.1

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

$- 3 x \cdot x^{2} + 6 x + x^{4} - 6 x^{2} + 8 = 0$

Step 9.2.2

Factor $- 3 x$ out of $6 x$ .

$- 3 x \cdot x^{2} - 3 x \cdot - 2 + x^{4} - 6 x^{2} + 8 = 0$

Step 9.2.3

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

$- 3 x (x^{2} - 2) + x^{4} - 6 x^{2} + 8 = 0$

Step 9.3

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

$- 3 x (x^{2} - 2) + {(x^{2})}^{2} - 6 x^{2} + 8 = 0$

Step 9.4

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

$- 3 x (x^{2} - 2) + u^{2} - 6 u + 8 = 0$

Step 9.5

Factor $u^{2} - 6 u + 8$ using the AC method.

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Step 9.5.1

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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 9.5.2

Write the factored form using these integers.

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

Step 9.6

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

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

Step 9.7

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

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

Step 9.8

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 $b = 2$ .

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

Step 9.9

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

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Step 9.9.1

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

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

Step 9.9.2

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

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

Step 9.9.3

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

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

Step 9.10

Expand $(x + 2) (x - 2)$ using the FOIL Method.

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Step 9.10.1

Apply the distributive property.

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

Step 9.10.2

Apply the distributive property.

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

Step 9.10.3

Apply the distributive property.

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

Step 9.11

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 9.11.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

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

Step 9.11.2

Add $- 2 x$ and $2 x$ .

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

Step 9.11.3

Add $x \cdot x$ and $0$ .

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

Step 9.12

Simplify each term.

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Step 9.12.1

Multiply $x$ by $x$ .

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

Step 9.12.2

Multiply $2$ by $- 2$ .

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

Step 9.13

Reorder terms.

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

Step 9.14

Factor.

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Step 9.14.1

Factor $x^{2} - 3 x - 4$ using the AC method.

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Step 9.14.1.1

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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 9.14.1.2

Write the factored form using these integers.

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

Step 9.14.2

Remove unnecessary parentheses.

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

Step 10

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

$x^{2} - 2 = 0$

$x - 4 = 0$

$x + 1 = 0$

Step 11

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

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Step 11.1

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

$x^{2} - 2 = 0$

Step 11.2

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

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Step 11.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 11.2.2

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

$x = \pm \sqrt{2}$

Step 11.2.3

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

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Step 11.2.3.1

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

$x = \sqrt{2}$

Step 11.2.3.2

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

$x = - \sqrt{2}$

Step 11.2.3.3

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

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

Step 12

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 12.2

Add $4$ to both sides of the equation.

$x = 4$

Step 13

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

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Step 13.1

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

$x + 1 = 0$

Step 13.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 14

The final solution is all the values that make $(x^{2} - 2) (x - 4) (x + 1) = 0$ true.

$x = \sqrt{2}, - \sqrt{2}, 4, - 1$

Step 15

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{2}, - \sqrt{2}, 4, - 1$

Decimal Form:

$x = 1.41421356 \dots, - 1.41421356 \dots, 4, - 1$

Step 16