Precalculus Examples

$x^{5} - 8 x^{4} + 22 x^{3} - 26 x^{2} + 21 x - 18 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $x^{5} - 8 x^{4} + 22 x^{3} - 26 x^{2} + 21 x - 18$ using the rational roots test.

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Step 1.1.1

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.

$p = \pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6$

$q = \pm 1$

Step 1.1.2

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

$\pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6$

Step 1.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 1.1.3.1

Substitute $2$ into the polynomial.

$2^{5} - 8 \cdot 2^{4} + 22 \cdot 2^{3} - 26 \cdot 2^{2} + 21 \cdot 2 - 18$

Step 1.1.3.2

Raise $2$ to the power of $5$ .

$32 - 8 \cdot 2^{4} + 22 \cdot 2^{3} - 26 \cdot 2^{2} + 21 \cdot 2 - 18$

Step 1.1.3.3

Raise $2$ to the power of $4$ .

$32 - 8 \cdot 16 + 22 \cdot 2^{3} - 26 \cdot 2^{2} + 21 \cdot 2 - 18$

Step 1.1.3.4

Multiply $- 8$ by $16$ .

$32 - 128 + 22 \cdot 2^{3} - 26 \cdot 2^{2} + 21 \cdot 2 - 18$

Step 1.1.3.5

Subtract $128$ from $32$ .

$- 96 + 22 \cdot 2^{3} - 26 \cdot 2^{2} + 21 \cdot 2 - 18$

Step 1.1.3.6

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

$- 96 + 22 \cdot 8 - 26 \cdot 2^{2} + 21 \cdot 2 - 18$

Step 1.1.3.7

Multiply $22$ by $8$ .

$- 96 + 176 - 26 \cdot 2^{2} + 21 \cdot 2 - 18$

Step 1.1.3.8

Add $- 96$ and $176$ .

$80 - 26 \cdot 2^{2} + 21 \cdot 2 - 18$

Step 1.1.3.9

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

$80 - 26 \cdot 4 + 21 \cdot 2 - 18$

Step 1.1.3.10

Multiply $- 26$ by $4$ .

$80 - 104 + 21 \cdot 2 - 18$

Step 1.1.3.11

Subtract $104$ from $80$ .

$- 24 + 21 \cdot 2 - 18$

Step 1.1.3.12

Multiply $21$ by $2$ .

$- 24 + 42 - 18$

Step 1.1.3.13

Add $- 24$ and $42$ .

$18 - 18$

Step 1.1.3.14

Subtract $18$ from $18$ .

$0$

Step 1.1.4

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.

$\frac{x^{5} - 8 x^{4} + 22 x^{3} - 26 x^{2} + 21 x - 18}{x - 2}$

Step 1.1.5

Divide $x^{5} - 8 x^{4} + 22 x^{3} - 26 x^{2} + 21 x - 18$ by $x - 2$ .

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Step 1.1.5.1

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

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

Step 1.1.5.2

Divide the highest order term in the dividend $x^{5}$ by the highest order term in divisor $x$ .

					$x^{4}$
	$x$	-	$2$		$x^{5}$	-	$8 x^{4}$	+	$22 x^{3}$	-	$26 x^{2}$	+	$21 x$	-	$18$

Step 1.1.5.3

Multiply the new quotient term by the divisor.

$x^{4}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

Step 1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{5} - 2 x^{4}$

$x^{4}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

Step 1.1.5.5

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

$x^{4}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

Step 1.1.5.6

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

$x^{4}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

Step 1.1.5.7

Divide the highest order term in the dividend $- 6 x^{4}$ by the highest order term in divisor $x$ .

$x^{4}$

$6 x^{3}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

Step 1.1.5.8

Multiply the new quotient term by the divisor.

$x^{4}$

$6 x^{3}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

Step 1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{4} + 12 x^{3}$

$x^{4}$

$6 x^{3}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

Step 1.1.5.10

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

$x^{4}$

$6 x^{3}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

Step 1.1.5.11

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

$x^{4}$

$6 x^{3}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

Step 1.1.5.12

Divide the highest order term in the dividend $10 x^{3}$ by the highest order term in divisor $x$ .

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

Step 1.1.5.13

Multiply the new quotient term by the divisor.

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

Step 1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{3} - 20 x^{2}$

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

Step 1.1.5.15

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

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

$6 x^{2}$

Step 1.1.5.16

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

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

$6 x^{2}$

$21 x$

Step 1.1.5.17

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $x$ .

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$6 x$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

$6 x^{2}$

$21 x$

Step 1.1.5.18

Multiply the new quotient term by the divisor.

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$6 x$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

$6 x^{2}$

$21 x$

$6 x^{2}$

$12 x$

Step 1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} + 12 x$

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$6 x$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

$6 x^{2}$

$21 x$

$6 x^{2}$

$12 x$

Step 1.1.5.20

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

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$6 x$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

$6 x^{2}$

$21 x$

$6 x^{2}$

$12 x$

$9 x$

Step 1.1.5.21

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

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$6 x$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

$6 x^{2}$

$21 x$

$6 x^{2}$

$12 x$

$9 x$

$18$

Step 1.1.5.22

Divide the highest order term in the dividend $9 x$ by the highest order term in divisor $x$ .

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$6 x$

$9$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

$6 x^{2}$

$21 x$

$6 x^{2}$

$12 x$

$9 x$

$18$

Step 1.1.5.23

Multiply the new quotient term by the divisor.

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$6 x$

$9$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

$6 x^{2}$

$21 x$

$6 x^{2}$

$12 x$

$9 x$

$18$

$9 x$

$18$

Step 1.1.5.24

The expression needs to be subtracted from the dividend, so change all the signs in $9 x - 18$

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$6 x$

$9$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

$6 x^{2}$

$21 x$

$6 x^{2}$

$12 x$

$9 x$

$18$

$9 x$

$18$

Step 1.1.5.25

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

$x^{4}$

$6 x^{3}$

$10 x^{2}$

$6 x$

$9$

$x$

$2$

$x^{5}$

$8 x^{4}$

$22 x^{3}$

$26 x^{2}$

$21 x$

$18$

$x^{5}$

$2 x^{4}$

$6 x^{4}$

$22 x^{3}$

$6 x^{4}$

$12 x^{3}$

$10 x^{3}$

$26 x^{2}$

$10 x^{3}$

$20 x^{2}$

$6 x^{2}$

$21 x$

$6 x^{2}$

$12 x$

$9 x$

$18$

$9 x$

$18$

$0$

Step 1.1.5.26

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

$x^{4} - 6 x^{3} + 10 x^{2} - 6 x + 9$

Step 1.1.6

Write $x^{5} - 8 x^{4} + 22 x^{3} - 26 x^{2} + 21 x - 18$ as a set of factors.

$(x - 2) (x^{4} - 6 x^{3} + 10 x^{2} - 6 x + 9) = 0$

Step 1.2

Regroup terms.

$(x - 2) (- 6 x^{3} - 6 x + x^{4} + 10 x^{2} + 9) = 0$

Step 1.3

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

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Step 1.3.1

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

$(x - 2) (- 6 x \cdot x^{2} - 6 x + x^{4} + 10 x^{2} + 9) = 0$

Step 1.3.2

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

$(x - 2) (- 6 x \cdot x^{2} - 6 x \cdot 1 + x^{4} + 10 x^{2} + 9) = 0$

Step 1.3.3

Factor $- 6 x$ out of $- 6 x (x^{2}) - 6 x (1)$ .

$(x - 2) (- 6 x (x^{2} + 1) + x^{4} + 10 x^{2} + 9) = 0$

Step 1.4

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

$(x - 2) (- 6 x (x^{2} + 1) + {(x^{2})}^{2} + 10 x^{2} + 9) = 0$

Step 1.5

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

$(x - 2) (- 6 x (x^{2} + 1) + u^{2} + 10 u + 9) = 0$

Step 1.6

Factor $u^{2} + 10 u + 9$ using the AC method.

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Step 1.6.1

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 $9$ and whose sum is $10$ .

$1, 9$

Step 1.6.2

Write the factored form using these integers.

$(x - 2) (- 6 x (x^{2} + 1) + (u + 1) (u + 9)) = 0$

Step 1.7

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

$(x - 2) (- 6 x (x^{2} + 1) + (x^{2} + 1) (x^{2} + 9)) = 0$

Step 1.8

Factor $x^{2} + 1$ out of $- 6 x (x^{2} + 1) + (x^{2} + 1) (x^{2} + 9)$ .

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Step 1.8.1

Factor $x^{2} + 1$ out of $- 6 x (x^{2} + 1)$ .

$(x - 2) ((x^{2} + 1) (- 6 x) + (x^{2} + 1) (x^{2} + 9)) = 0$

Step 1.8.2

Factor $x^{2} + 1$ out of $(x^{2} + 1) (- 6 x) + (x^{2} + 1) (x^{2} + 9)$ .

$(x - 2) ((x^{2} + 1) (- 6 x + x^{2} + 9)) = 0$

Step 1.9

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x - 2) ((x^{2} + 1) (- 6 u + u^{2} + 9)) = 0$

Step 1.10

Factor using the perfect square rule.

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Step 1.10.1

Rearrange terms.

$(x - 2) ((x^{2} + 1) (u^{2} - 6 u + 9)) = 0$

Step 1.10.2

Rewrite $9$ as $3^{2}$ .

$(x - 2) ((x^{2} + 1) (u^{2} - 6 u + 3^{2})) = 0$

Step 1.10.3

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

$6 u = 2 \cdot u \cdot 3$

Step 1.10.4

Rewrite the polynomial.

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

Step 1.10.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 3$ .

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

Step 1.11

Replace all occurrences of $u$ with $x$ .

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Step 1.11.1

Replace all occurrences of $u$ with $x$ .

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

Step 1.11.2

Remove unnecessary parentheses.

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

Step 2

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

$x - 2 = 0$

$x^{2} + 1 = 0$

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

Step 3

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

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Step 3.1

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

$x - 2 = 0$

Step 3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 4.2

Solve $x^{2} + 1 = 0$ for $x$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 4.2.2

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

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

Step 4.2.3

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

$x = \pm i$

Step 4.2.4

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

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Step 4.2.4.1

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

$x = i$

Step 4.2.4.2

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

$x = - i$

Step 4.2.4.3

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

$x = i, - i$

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 5.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6

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

$x = 2, i, - i, 3$