Precalculus Examples

$4 x^{4} - 22 x^{3} + 36 x^{2} - 44 x + 56$

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.

$p = \pm 1, \pm 2, \pm 4, \pm 7, \pm 8, \pm 14, \pm 28, \pm 56$

$q = \pm 1, \pm 2, \pm 4$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2, \pm 4, \pm 7, \pm \frac{7}{2}, \pm \frac{7}{4}, \pm 8, \pm 14, \pm 28, \pm 56$

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.

$4 {(2)}^{4} - 22 {(2)}^{3} + 36 {(2)}^{2} - 44 \cdot 2 + 56$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 2$ 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 $2$ to the power of $4$ .

$4 \cdot 16 - 22 {(2)}^{3} + 36 {(2)}^{2} - 44 \cdot 2 + 56$

Step 4.1.2

Multiply $4$ by $16$ .

$64 - 22 {(2)}^{3} + 36 {(2)}^{2} - 44 \cdot 2 + 56$

Step 4.1.3

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

$64 - 22 \cdot 8 + 36 {(2)}^{2} - 44 \cdot 2 + 56$

Step 4.1.4

Multiply $- 22$ by $8$ .

$64 - 176 + 36 {(2)}^{2} - 44 \cdot 2 + 56$

Step 4.1.5

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

$64 - 176 + 36 \cdot 4 - 44 \cdot 2 + 56$

Step 4.1.6

Multiply $36$ by $4$ .

$64 - 176 + 144 - 44 \cdot 2 + 56$

Step 4.1.7

Multiply $- 44$ by $2$ .

$64 - 176 + 144 - 88 + 56$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $176$ from $64$ .

$- 112 + 144 - 88 + 56$

Step 4.2.2

Add $- 112$ and $144$ .

$32 - 88 + 56$

Step 4.2.3

Subtract $88$ from $32$ .

$- 56 + 56$

Step 4.2.4

Add $- 56$ and $56$ .

$0$

Step 5

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{4} - 22 x^{3} + 36 x^{2} - 44 x + 56}{x - 2}$

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.

$2$	$4$	$- 22$	$36$	$- 44$	$56$

Step 6.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$2$	$4$	$- 22$	$36$	$- 44$	$56$

	$4$

Step 6.3

Multiply the newest entry in the result $(4)$ by the divisor $(2)$ and place the result of $(8)$ under the next term in the dividend $(- 22)$ .

$2$	$4$	$- 22$	$36$	$- 44$	$56$
		$8$
	$4$

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.

$2$	$4$	$- 22$	$36$	$- 44$	$56$
		$8$
	$4$	$- 14$

Step 6.5

Multiply the newest entry in the result $(- 14)$ by the divisor $(2)$ and place the result of $(- 28)$ under the next term in the dividend $(36)$ .

$2$	$4$	$- 22$	$36$	$- 44$	$56$
		$8$	$- 28$
	$4$	$- 14$

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.

$2$	$4$	$- 22$	$36$	$- 44$	$56$
		$8$	$- 28$
	$4$	$- 14$	$8$

Step 6.7

Multiply the newest entry in the result $(8)$ by the divisor $(2)$ and place the result of $(16)$ under the next term in the dividend $(- 44)$ .

$2$	$4$	$- 22$	$36$	$- 44$	$56$
		$8$	$- 28$	$16$
	$4$	$- 14$	$8$

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.

$2$	$4$	$- 22$	$36$	$- 44$	$56$
		$8$	$- 28$	$16$
	$4$	$- 14$	$8$	$- 28$

Step 6.9

Multiply the newest entry in the result $(- 28)$ by the divisor $(2)$ and place the result of $(- 56)$ under the next term in the dividend $(56)$ .

$2$	$4$	$- 22$	$36$	$- 44$	$56$
		$8$	$- 28$	$16$	$- 56$
	$4$	$- 14$	$8$	$- 28$

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.

$2$	$4$	$- 22$	$36$	$- 44$	$56$
		$8$	$- 28$	$16$	$- 56$
	$4$	$- 14$	$8$	$- 28$	$0$

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.

$4 x^{3} + - 14 x^{2} + (8) x - 28$

Step 6.12

Simplify the quotient polynomial.

$4 x^{3} - 14 x^{2} + 8 x - 28$

Step 7

Factor $2$ out of $4 x^{3} - 14 x^{2} + 8 x - 28$ .

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Step 7.1

Factor $2$ out of $4 x^{3}$ .

$(x - 2) (2 (2 x^{3}) - 14 x^{2} + 8 x - 28)$

Step 7.2

Factor $2$ out of $- 14 x^{2}$ .

$(x - 2) (2 (2 x^{3}) + 2 (- 7 x^{2}) + 8 x - 28)$

Step 7.3

Factor $2$ out of $8 x$ .

$(x - 2) (2 (2 x^{3}) + 2 (- 7 x^{2}) + 2 (4 x) - 28)$

Step 7.4

Factor $2$ out of $- 28$ .

$(x - 2) (2 (2 x^{3}) + 2 (- 7 x^{2}) + 2 (4 x) + 2 (- 14))$

Step 7.5

Factor $2$ out of $2 (2 x^{3}) + 2 (- 7 x^{2})$ .

$(x - 2) (2 (2 x^{3} - 7 x^{2}) + 2 (4 x) + 2 (- 14))$

Step 7.6

Factor $2$ out of $2 (2 x^{3} - 7 x^{2}) + 2 (4 x)$ .

$(x - 2) (2 (2 x^{3} - 7 x^{2} + 4 x) + 2 (- 14))$

Step 7.7

Factor $2$ out of $2 (2 x^{3} - 7 x^{2} + 4 x) + 2 (- 14)$ .

$(x - 2) (2 (2 x^{3} - 7 x^{2} + 4 x - 14))$

Step 8

Factor out the greatest common factor from each group.

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Step 8.1

Group the first two terms and the last two terms.

$(x - 2) (2 ((2 x^{3} - 7 x^{2}) + 4 x - 14))$

Step 8.2

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

$(x - 2) (2 (x^{2} (2 x - 7) + 2 (2 x - 7)))$

Step 9

Factor.

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Step 9.1

Factor the polynomial by factoring out the greatest common factor, $2 x - 7$ .

$(x - 2) (2 ((2 x - 7) (x^{2} + 2)))$

Step 9.2

Remove unnecessary parentheses.

$(x - 2) (2 (2 x - 7) (x^{2} + 2))$

Step 10

Factor the left side of the equation.

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Step 10.1

Factor $2$ out of $4 x^{4} - 22 x^{3} + 36 x^{2} - 44 x + 56$ .

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Step 10.1.1

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

$2 (2 x^{4}) - 22 x^{3} + 36 x^{2} - 44 x + 56 = 0$

Step 10.1.2

Factor $2$ out of $- 22 x^{3}$ .

$2 (2 x^{4}) + 2 (- 11 x^{3}) + 36 x^{2} - 44 x + 56 = 0$

Step 10.1.3

Factor $2$ out of $36 x^{2}$ .

$2 (2 x^{4}) + 2 (- 11 x^{3}) + 2 (18 x^{2}) - 44 x + 56 = 0$

Step 10.1.4

Factor $2$ out of $- 44 x$ .

$2 (2 x^{4}) + 2 (- 11 x^{3}) + 2 (18 x^{2}) + 2 (- 22 x) + 56 = 0$

Step 10.1.5

Factor $2$ out of $56$ .

$2 (2 x^{4}) + 2 (- 11 x^{3}) + 2 (18 x^{2}) + 2 (- 22 x) + 2 (28) = 0$

Step 10.1.6

Factor $2$ out of $2 (2 x^{4}) + 2 (- 11 x^{3})$ .

$2 (2 x^{4} - 11 x^{3}) + 2 (18 x^{2}) + 2 (- 22 x) + 2 (28) = 0$

Step 10.1.7

Factor $2$ out of $2 (2 x^{4} - 11 x^{3}) + 2 (18 x^{2})$ .

$2 (2 x^{4} - 11 x^{3} + 18 x^{2}) + 2 (- 22 x) + 2 (28) = 0$

Step 10.1.8

Factor $2$ out of $2 (2 x^{4} - 11 x^{3} + 18 x^{2}) + 2 (- 22 x)$ .

$2 (2 x^{4} - 11 x^{3} + 18 x^{2} - 22 x) + 2 (28) = 0$

Step 10.1.9

Factor $2$ out of $2 (2 x^{4} - 11 x^{3} + 18 x^{2} - 22 x) + 2 (28)$ .

$2 (2 x^{4} - 11 x^{3} + 18 x^{2} - 22 x + 28) = 0$

Step 10.2

Regroup terms.

$2 (- 11 x^{3} - 22 x + 2 x^{4} + 18 x^{2} + 28) = 0$

Step 10.3

Factor $- 11 x$ out of $- 11 x^{3} - 22 x$ .

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Step 10.3.1

Factor $- 11 x$ out of $- 11 x^{3}$ .

$2 (- 11 x \cdot x^{2} - 22 x + 2 x^{4} + 18 x^{2} + 28) = 0$

Step 10.3.2

Factor $- 11 x$ out of $- 22 x$ .

$2 (- 11 x \cdot x^{2} - 11 x \cdot 2 + 2 x^{4} + 18 x^{2} + 28) = 0$

Step 10.3.3

Factor $- 11 x$ out of $- 11 x (x^{2}) - 11 x (2)$ .

$2 (- 11 x (x^{2} + 2) + 2 x^{4} + 18 x^{2} + 28) = 0$

Step 10.4

Factor $2$ out of $2 x^{4} + 18 x^{2} + 28$ .

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Step 10.4.1

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

$2 (- 11 x (x^{2} + 2) + 2 (x^{4}) + 18 x^{2} + 28) = 0$

Step 10.4.2

Factor $2$ out of $18 x^{2}$ .

$2 (- 11 x (x^{2} + 2) + 2 (x^{4}) + 2 (9 x^{2}) + 28) = 0$

Step 10.4.3

Factor $2$ out of $28$ .

$2 (- 11 x (x^{2} + 2) + 2 x^{4} + 2 (9 x^{2}) + 2 \cdot 14) = 0$

Step 10.4.4

Factor $2$ out of $2 x^{4} + 2 (9 x^{2})$ .

$2 (- 11 x (x^{2} + 2) + 2 (x^{4} + 9 x^{2}) + 2 \cdot 14) = 0$

Step 10.4.5

Factor $2$ out of $2 (x^{4} + 9 x^{2}) + 2 \cdot 14$ .

$2 (- 11 x (x^{2} + 2) + 2 (x^{4} + 9 x^{2} + 14)) = 0$

Step 10.5

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

$2 (- 11 x (x^{2} + 2) + 2 ({(x^{2})}^{2} + 9 x^{2} + 14)) = 0$

Step 10.6

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

$2 (- 11 x (x^{2} + 2) + 2 (u^{2} + 9 u + 14)) = 0$

Step 10.7

Factor $u^{2} + 9 u + 14$ using the AC method.

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Step 10.7.1

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 $14$ and whose sum is $9$ .

$2, 7$

Step 10.7.2

Write the factored form using these integers.

$2 (- 11 x (x^{2} + 2) + 2 ((u + 2) (u + 7))) = 0$

Step 10.8

Factor.

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Step 10.8.1

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

$2 (- 11 x (x^{2} + 2) + 2 ((x^{2} + 2) (x^{2} + 7))) = 0$

Step 10.8.2

Remove unnecessary parentheses.

$2 (- 11 x (x^{2} + 2) + 2 (x^{2} + 2) (x^{2} + 7)) = 0$

Step 10.9

Factor $x^{2} + 2$ out of $- 11 x (x^{2} + 2) + 2 (x^{2} + 2) (x^{2} + 7)$ .

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Step 10.9.1

Factor $x^{2} + 2$ out of $- 11 x (x^{2} + 2)$ .

$2 ((x^{2} + 2) (- 11 x) + 2 (x^{2} + 2) (x^{2} + 7)) = 0$

Step 10.9.2

Factor $x^{2} + 2$ out of $2 (x^{2} + 2) (x^{2} + 7)$ .

$2 ((x^{2} + 2) (- 11 x) + (x^{2} + 2) (2 (x^{2} + 7))) = 0$

Step 10.9.3

Factor $x^{2} + 2$ out of $(x^{2} + 2) (- 11 x) + (x^{2} + 2) (2 (x^{2} + 7))$ .

$2 ((x^{2} + 2) (- 11 x + 2 (x^{2} + 7))) = 0$

Step 10.10

Apply the distributive property.

$2 ((x^{2} + 2) (- 11 x + 2 x^{2} + 2 \cdot 7)) = 0$

Step 10.11

Multiply $2$ by $7$ .

$2 ((x^{2} + 2) (- 11 x + 2 x^{2} + 14)) = 0$

Step 10.12

Reorder terms.

$2 ((x^{2} + 2) (2 x^{2} - 11 x + 14)) = 0$

Step 10.13

Factor.

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Step 10.13.1

Factor.

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Step 10.13.1.1

Factor by grouping.

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Step 10.13.1.1.1

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 $a \cdot c = 2 \cdot 14 = 28$ and whose sum is $b = - 11$ .

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Step 10.13.1.1.1.1

Factor $- 11$ out of $- 11 x$ .

$2 ((x^{2} + 2) (2 x^{2} - 11 x + 14)) = 0$

Step 10.13.1.1.1.2

Rewrite $- 11$ as $- 4$ plus $- 7$

$2 ((x^{2} + 2) (2 x^{2} + (- 4 - 7) x + 14)) = 0$

Step 10.13.1.1.1.3

Apply the distributive property.

$2 ((x^{2} + 2) (2 x^{2} - 4 x - 7 x + 14)) = 0$

Step 10.13.1.1.2

Factor out the greatest common factor from each group.

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Step 10.13.1.1.2.1

Group the first two terms and the last two terms.

$2 ((x^{2} + 2) ((2 x^{2} - 4 x) - 7 x + 14)) = 0$

Step 10.13.1.1.2.2

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

$2 ((x^{2} + 2) (2 x (x - 2) - 7 (x - 2))) = 0$

Step 10.13.1.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$2 ((x^{2} + 2) ((x - 2) (2 x - 7))) = 0$

Step 10.13.1.2

Remove unnecessary parentheses.

$2 ((x^{2} + 2) (x - 2) (2 x - 7)) = 0$

Step 10.13.2

Remove unnecessary parentheses.

$2 (x^{2} + 2) (x - 2) (2 x - 7) = 0$

Step 11

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

$x^{2} + 2 = 0$

$x - 2 = 0$

$2 x - 7 = 0$

Step 12

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

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Step 12.1

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

$x^{2} + 2 = 0$

Step 12.2

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

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Step 12.2.1

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

Step 12.2.2

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

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

Step 12.2.3

Simplify $\pm \sqrt{- 2}$ .

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Step 12.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 12.2.3.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 12.2.3.3

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

$x = \pm i \sqrt{2}$

Step 12.2.4

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

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Step 12.2.4.1

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

$x = i \sqrt{2}$

Step 12.2.4.2

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

$x = - i \sqrt{2}$

Step 12.2.4.3

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

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

Step 13

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

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Step 13.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 13.2

Add $2$ to both sides of the equation.

$x = 2$

Step 14

Set $2 x - 7$ equal to $0$ and solve for $x$ .

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Step 14.1

Set $2 x - 7$ equal to $0$ .

$2 x - 7 = 0$

Step 14.2

Solve $2 x - 7 = 0$ for $x$ .

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Step 14.2.1

Add $7$ to both sides of the equation.

$2 x = 7$

Step 14.2.2

Divide each term in $2 x = 7$ by $2$ and simplify.

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Step 14.2.2.1

Divide each term in $2 x = 7$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2}$

Step 14.2.2.2

Simplify the left side.

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Step 14.2.2.2.1

Cancel the common factor of $2$ .

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Step 14.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2}$

Step 14.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{2}$

Step 15

The final solution is all the values that make $2 (x^{2} + 2) (x - 2) (2 x - 7) = 0$ true.

$x = i \sqrt{2}, - i \sqrt{2}, 2, \frac{7}{2}$

Step 16