Precalculus Examples

$2 x^{5} - 13 x^{4} + 24 x^{3} - 34 x^{2} + 150 x - 225$

Step 1

Set $2 x^{5} - 13 x^{4} + 24 x^{3} - 34 x^{2} + 150 x - 225$ equal to $0$ .

$2 x^{5} - 13 x^{4} + 24 x^{3} - 34 x^{2} + 150 x - 225 = 0$

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

Factor $2 x^{5} - 13 x^{4} + 24 x^{3} - 34 x^{2} + 150 x - 225$ using the rational roots test.

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Step 2.1.1.1

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.

$p = \pm 1, \pm 225, \pm 3, \pm 75, \pm 5, \pm 45, \pm 9, \pm 25, \pm 15$

$q = \pm 1, \pm 2$

Step 2.1.1.2

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

$\pm 1, \pm 0.5, \pm 225, \pm 112.5, \pm 3, \pm 1.5, \pm 75, \pm 37.5, \pm 5, \pm 2.5, \pm 45, \pm 22.5, \pm 9, \pm 4.5, \pm 25, \pm 12.5, \pm 15, \pm 7.5$

Step 2.1.1.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 2.1.1.3.1

Substitute $3$ into the polynomial.

$2 \cdot 3^{5} - 13 \cdot 3^{4} + 24 \cdot 3^{3} - 34 \cdot 3^{2} + 150 \cdot 3 - 225$

Step 2.1.1.3.2

Raise $3$ to the power of $5$ .

$2 \cdot 243 - 13 \cdot 3^{4} + 24 \cdot 3^{3} - 34 \cdot 3^{2} + 150 \cdot 3 - 225$

Step 2.1.1.3.3

Multiply $2$ by $243$ .

$486 - 13 \cdot 3^{4} + 24 \cdot 3^{3} - 34 \cdot 3^{2} + 150 \cdot 3 - 225$

Step 2.1.1.3.4

Raise $3$ to the power of $4$ .

$486 - 13 \cdot 81 + 24 \cdot 3^{3} - 34 \cdot 3^{2} + 150 \cdot 3 - 225$

Step 2.1.1.3.5

Multiply $- 13$ by $81$ .

$486 - 1053 + 24 \cdot 3^{3} - 34 \cdot 3^{2} + 150 \cdot 3 - 225$

Step 2.1.1.3.6

Subtract $1053$ from $486$ .

$- 567 + 24 \cdot 3^{3} - 34 \cdot 3^{2} + 150 \cdot 3 - 225$

Step 2.1.1.3.7

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

$- 567 + 24 \cdot 27 - 34 \cdot 3^{2} + 150 \cdot 3 - 225$

Step 2.1.1.3.8

Multiply $24$ by $27$ .

$- 567 + 648 - 34 \cdot 3^{2} + 150 \cdot 3 - 225$

Step 2.1.1.3.9

Add $- 567$ and $648$ .

$81 - 34 \cdot 3^{2} + 150 \cdot 3 - 225$

Step 2.1.1.3.10

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

$81 - 34 \cdot 9 + 150 \cdot 3 - 225$

Step 2.1.1.3.11

Multiply $- 34$ by $9$ .

$81 - 306 + 150 \cdot 3 - 225$

Step 2.1.1.3.12

Subtract $306$ from $81$ .

$- 225 + 150 \cdot 3 - 225$

Step 2.1.1.3.13

Multiply $150$ by $3$ .

$- 225 + 450 - 225$

Step 2.1.1.3.14

Add $- 225$ and $450$ .

$225 - 225$

Step 2.1.1.3.15

Subtract $225$ from $225$ .

$0$

Step 2.1.1.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{5} - 13 x^{4} + 24 x^{3} - 34 x^{2} + 150 x - 225}{x - 3}$

Step 2.1.1.5

Divide $2 x^{5} - 13 x^{4} + 24 x^{3} - 34 x^{2} + 150 x - 225$ by $x - 3$ .

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Step 2.1.1.5.1

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

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

Step 2.1.1.5.2

Divide the highest order term in the dividend $2 x^{5}$ by the highest order term in divisor $x$ .

					$2 x^{4}$
	$x$	-	$3$		$2 x^{5}$	-	$13 x^{4}$	+	$24 x^{3}$	-	$34 x^{2}$	+	$150 x$	-	$225$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

$2 x^{4}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

Step 2.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{5} - 6 x^{4}$

$2 x^{4}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

Step 2.1.1.5.5

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

$2 x^{4}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

Step 2.1.1.5.6

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

$2 x^{4}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

Step 2.1.1.5.7

Divide the highest order term in the dividend $- 7 x^{4}$ by the highest order term in divisor $x$ .

$2 x^{4}$

$7 x^{3}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

$2 x^{4}$

$7 x^{3}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

Step 2.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{4} + 21 x^{3}$

$2 x^{4}$

$7 x^{3}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

Step 2.1.1.5.10

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

$2 x^{4}$

$7 x^{3}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

Step 2.1.1.5.11

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

$2 x^{4}$

$7 x^{3}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

Step 2.1.1.5.12

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

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} - 9 x^{2}$

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

Step 2.1.1.5.15

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

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

$25 x^{2}$

Step 2.1.1.5.16

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

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

$25 x^{2}$

$150 x$

Step 2.1.1.5.17

Divide the highest order term in the dividend $- 25 x^{2}$ by the highest order term in divisor $x$ .

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$25 x$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

$25 x^{2}$

$150 x$

Step 2.1.1.5.18

Multiply the new quotient term by the divisor.

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$25 x$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

$25 x^{2}$

$150 x$

$25 x^{2}$

$75 x$

Step 2.1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 25 x^{2} + 75 x$

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$25 x$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

$25 x^{2}$

$150 x$

$25 x^{2}$

$75 x$

Step 2.1.1.5.20

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

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$25 x$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

$25 x^{2}$

$150 x$

$25 x^{2}$

$75 x$

Step 2.1.1.5.21

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

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$25 x$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

$25 x^{2}$

$150 x$

$25 x^{2}$

$75 x$

$225$

Step 2.1.1.5.22

Divide the highest order term in the dividend $75 x$ by the highest order term in divisor $x$ .

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$25 x$

$75$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

$25 x^{2}$

$150 x$

$25 x^{2}$

$75 x$

$225$

Step 2.1.1.5.23

Multiply the new quotient term by the divisor.

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$25 x$

$75$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

$25 x^{2}$

$150 x$

$25 x^{2}$

$75 x$

$225$

$75 x$

$225$

Step 2.1.1.5.24

The expression needs to be subtracted from the dividend, so change all the signs in $75 x - 225$

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$25 x$

$75$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

$25 x^{2}$

$150 x$

$25 x^{2}$

$75 x$

$225$

$75 x$

$225$

Step 2.1.1.5.25

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

$2 x^{4}$

$7 x^{3}$

$3 x^{2}$

$25 x$

$75$

$x$

$3$

$2 x^{5}$

$13 x^{4}$

$24 x^{3}$

$34 x^{2}$

$150 x$

$225$

$2 x^{5}$

$6 x^{4}$

$7 x^{4}$

$24 x^{3}$

$7 x^{4}$

$21 x^{3}$

$3 x^{3}$

$34 x^{2}$

$3 x^{3}$

$9 x^{2}$

$25 x^{2}$

$150 x$

$25 x^{2}$

$75 x$

$225$

$75 x$

$225$

$0$

Step 2.1.1.5.26

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

$2 x^{4} - 7 x^{3} + 3 x^{2} - 25 x + 75$

Step 2.1.1.6

Write $2 x^{5} - 13 x^{4} + 24 x^{3} - 34 x^{2} + 150 x - 225$ as a set of factors.

$(x - 3) (2 x^{4} - 7 x^{3} + 3 x^{2} - 25 x + 75) = 0$

Step 2.1.2

Regroup terms.

$(x - 3) (- 25 x + 75 + 2 x^{4} - 7 x^{3} + 3 x^{2}) = 0$

Step 2.1.3

Factor $- 25$ out of $- 25 x + 75$ .

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Step 2.1.3.1

Factor $- 25$ out of $- 25 x$ .

$(x - 3) (- 25 x + 75 + 2 x^{4} - 7 x^{3} + 3 x^{2}) = 0$

Step 2.1.3.2

Factor $- 25$ out of $75$ .

$(x - 3) (- 25 x - 25 \cdot - 3 + 2 x^{4} - 7 x^{3} + 3 x^{2}) = 0$

Step 2.1.3.3

Factor $- 25$ out of $- 25 (x) - 25 (- 3)$ .

$(x - 3) (- 25 (x - 3) + 2 x^{4} - 7 x^{3} + 3 x^{2}) = 0$

Step 2.1.4

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

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Step 2.1.4.1

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

$(x - 3) (- 25 (x - 3) + x^{2} (2 x^{2}) - 7 x^{3} + 3 x^{2}) = 0$

Step 2.1.4.2

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

$(x - 3) (- 25 (x - 3) + x^{2} (2 x^{2}) + x^{2} (- 7 x) + 3 x^{2}) = 0$

Step 2.1.4.3

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

$(x - 3) (- 25 (x - 3) + x^{2} (2 x^{2}) + x^{2} (- 7 x) + x^{2} \cdot 3) = 0$

Step 2.1.4.4

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

$(x - 3) (- 25 (x - 3) + x^{2} (2 x^{2} - 7 x) + x^{2} \cdot 3) = 0$

Step 2.1.4.5

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

$(x - 3) (- 25 (x - 3) + x^{2} (2 x^{2} - 7 x + 3)) = 0$

Step 2.1.5

Factor.

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Step 2.1.5.1

Factor by grouping.

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Step 2.1.5.1.1

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = - 7$ .

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Step 2.1.5.1.1.1

Factor $- 7$ out of $- 7 x$ .

$(x - 3) (- 25 (x - 3) + x^{2} (2 x^{2} - 7 x + 3)) = 0$

Step 2.1.5.1.1.2

Rewrite $- 7$ as $- 1$ plus $- 6$

$(x - 3) (- 25 (x - 3) + x^{2} (2 x^{2} + (- 1 - 6) x + 3)) = 0$

Step 2.1.5.1.1.3

Apply the distributive property.

$(x - 3) (- 25 (x - 3) + x^{2} (2 x^{2} - 1 x - 6 x + 3)) = 0$

Step 2.1.5.1.2

Factor out the greatest common factor from each group.

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Step 2.1.5.1.2.1

Group the first two terms and the last two terms.

$(x - 3) (- 25 (x - 3) + x^{2} ((2 x^{2} - 1 x) - 6 x + 3)) = 0$

Step 2.1.5.1.2.2

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

$(x - 3) (- 25 (x - 3) + x^{2} (x (2 x - 1) - 3 (2 x - 1))) = 0$

Step 2.1.5.1.3

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

$(x - 3) (- 25 (x - 3) + x^{2} ((2 x - 1) (x - 3))) = 0$

Step 2.1.5.2

Remove unnecessary parentheses.

$(x - 3) (- 25 (x - 3) + x^{2} (2 x - 1) (x - 3)) = 0$

Step 2.1.6

Factor $x - 3$ out of $- 25 (x - 3) + x^{2} (2 x - 1) (x - 3)$ .

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Step 2.1.6.1

Factor $x - 3$ out of $- 25 (x - 3)$ .

$(x - 3) ((x - 3) \cdot - 25 + x^{2} (2 x - 1) (x - 3)) = 0$

Step 2.1.6.2

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

$(x - 3) ((x - 3) \cdot - 25 + (x - 3) (x^{2} (2 x - 1))) = 0$

Step 2.1.6.3

Factor $x - 3$ out of $(x - 3) \cdot - 25 + (x - 3) (x^{2} (2 x - 1))$ .

$(x - 3) ((x - 3) (- 25 + x^{2} (2 x - 1))) = 0$

Step 2.1.7

Apply the distributive property.

$(x - 3) ((x - 3) (- 25 + x^{2} (2 x) + x^{2} \cdot - 1)) = 0$

Step 2.1.8

Rewrite using the commutative property of multiplication.

$(x - 3) ((x - 3) (- 25 + 2 x^{2} x + x^{2} \cdot - 1)) = 0$

Step 2.1.9

Move $- 1$ to the left of $x^{2}$ .

$(x - 3) ((x - 3) (- 25 + 2 x^{2} x - 1 \cdot x^{2})) = 0$

Step 2.1.10

Simplify each term.

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Step 2.1.10.1

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

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Step 2.1.10.1.1

Move $x$ .

$(x - 3) ((x - 3) (- 25 + 2 (x \cdot x^{2}) - 1 \cdot x^{2})) = 0$

Step 2.1.10.1.2

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

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Step 2.1.10.1.2.1

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

$(x - 3) ((x - 3) (- 25 + 2 (x \cdot x^{2}) - 1 \cdot x^{2})) = 0$

Step 2.1.10.1.2.2

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

$(x - 3) ((x - 3) (- 25 + 2 x^{1 + 2} - 1 \cdot x^{2})) = 0$

Step 2.1.10.1.3

Add $1$ and $2$ .

$(x - 3) ((x - 3) (- 25 + 2 x^{3} - 1 \cdot x^{2})) = 0$

$(x - 3) ((x - 3) (- 25 + 2 x^{3} - 1 x^{2})) = 0$

Step 2.1.10.2

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

$(x - 3) ((x - 3) (- 25 + 2 x^{3} - x^{2})) = 0$

Step 2.1.11

Reorder terms.

$(x - 3) ((x - 3) (2 x^{3} - x^{2} - 25)) = 0$

Step 2.1.12

Factor.

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Step 2.1.12.1

Factor.

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Step 2.1.12.1.1

Factor $2 x^{3} - x^{2} - 25$ using the rational roots test.

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Step 2.1.12.1.1.1

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.

$p = \pm 1, \pm 25, \pm 5$

$q = \pm 1, \pm 2$

Step 2.1.12.1.1.2

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

$\pm 1, \pm 0.5, \pm 25, \pm 12.5, \pm 5, \pm 2.5$

Step 2.1.12.1.1.3

Substitute $2.5$ and simplify the expression. In this case, the expression is equal to $0$ so $2.5$ is a root of the polynomial.

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Step 2.1.12.1.1.3.1

Substitute $2.5$ into the polynomial.

$2 \cdot {2.5}^{3} - {2.5}^{2} - 25$

Step 2.1.12.1.1.3.2

Raise $2.5$ to the power of $3$ .

$2 \cdot 15.625 - {2.5}^{2} - 25$

Step 2.1.12.1.1.3.3

Multiply $2$ by $15.625$ .

$31.25 - {2.5}^{2} - 25$

Step 2.1.12.1.1.3.4

Raise $2.5$ to the power of $2$ .

$31.25 - 1 \cdot 6.25 - 25$

Step 2.1.12.1.1.3.5

Multiply $- 1$ by $6.25$ .

$31.25 - 6.25 - 25$

Step 2.1.12.1.1.3.6

Subtract $6.25$ from $31.25$ .

$25 - 25$

Step 2.1.12.1.1.3.7

Subtract $25$ from $25$ .

$0$

Step 2.1.12.1.1.4

Since $2.5$ is a known root, divide the polynomial by $2 x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} - x^{2} - 25}{2 x - 5}$

Step 2.1.12.1.1.5

Divide $2 x^{3} - x^{2} - 25$ by $2 x - 5$ .

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Step 2.1.12.1.1.5.1

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

$2 x$

$5$

$2 x^{3}$

$x^{2}$

$0 x$

$25$

Step 2.1.12.1.1.5.2

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

					$x^{2}$
	$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$

Step 2.1.12.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			+	$2 x^{3}$	-	$5 x^{2}$

Step 2.1.12.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} - 5 x^{2}$

				$x^{2}$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			-	$2 x^{3}$	+	$5 x^{2}$

Step 2.1.12.1.1.5.5

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

				$x^{2}$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			-	$2 x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$

Step 2.1.12.1.1.5.6

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

				$x^{2}$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			-	$2 x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$0 x$

Step 2.1.12.1.1.5.7

Divide the highest order term in the dividend $4 x^{2}$ by the highest order term in divisor $2 x$ .

				$x^{2}$	+	$2 x$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			-	$2 x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$0 x$

Step 2.1.12.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			-	$2 x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	-	$10 x$

Step 2.1.12.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{2} - 10 x$

				$x^{2}$	+	$2 x$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			-	$2 x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$0 x$
					-	$4 x^{2}$	+	$10 x$

Step 2.1.12.1.1.5.10

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

				$x^{2}$	+	$2 x$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			-	$2 x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$0 x$
					-	$4 x^{2}$	+	$10 x$
							+	$10 x$

Step 2.1.12.1.1.5.11

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

				$x^{2}$	+	$2 x$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			-	$2 x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$0 x$
					-	$4 x^{2}$	+	$10 x$
							+	$10 x$	-	$25$

Step 2.1.12.1.1.5.12

Divide the highest order term in the dividend $10 x$ by the highest order term in divisor $2 x$ .

				$x^{2}$	+	$2 x$	+	$5$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			-	$2 x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$0 x$
					-	$4 x^{2}$	+	$10 x$
							+	$10 x$	-	$25$

Step 2.1.12.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	+	$5$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			-	$2 x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$0 x$
					-	$4 x^{2}$	+	$10 x$
							+	$10 x$	-	$25$
							+	$10 x$	-	$25$

Step 2.1.12.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $10 x - 25$

				$x^{2}$	+	$2 x$	+	$5$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			-	$2 x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$0 x$
					-	$4 x^{2}$	+	$10 x$
							+	$10 x$	-	$25$
							-	$10 x$	+	$25$

Step 2.1.12.1.1.5.15

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

				$x^{2}$	+	$2 x$	+	$5$
$2 x$	-	$5$		$2 x^{3}$	-	$x^{2}$	+	$0 x$	-	$25$
			-	$2 x^{3}$	+	$5 x^{2}$
					+	$4 x^{2}$	+	$0 x$
					-	$4 x^{2}$	+	$10 x$
							+	$10 x$	-	$25$
							-	$10 x$	+	$25$
										$0$

Step 2.1.12.1.1.5.16

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

$x^{2} + 2 x + 5$

Step 2.1.12.1.1.6

Write $2 x^{3} - x^{2} - 25$ as a set of factors.

$(x - 3) ((x - 3) ((2 x - 5) (x^{2} + 2 x + 5))) = 0$

Step 2.1.12.1.2

Remove unnecessary parentheses.

$(x - 3) ((x - 3) (2 x - 5) (x^{2} + 2 x + 5)) = 0$

Step 2.1.12.2

Remove unnecessary parentheses.

$(x - 3) (x - 3) (2 x - 5) (x^{2} + 2 x + 5) = 0$

Step 2.1.13

Combine like factors.

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Step 2.1.13.1

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

$(x - 3) (x - 3) (2 x - 5) (x^{2} + 2 x + 5) = 0$

Step 2.1.13.2

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

$(x - 3) (x - 3) (2 x - 5) (x^{2} + 2 x + 5) = 0$

Step 2.1.13.3

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

${(x - 3)}^{1 + 1} (2 x - 5) (x^{2} + 2 x + 5) = 0$

Step 2.1.13.4

Add $1$ and $1$ .

${(x - 3)}^{2} (2 x - 5) (x^{2} + 2 x + 5) = 0$

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 $0$ .

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

$2 x - 5 = 0$

$x^{2} + 2 x + 5 = 0$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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Step 2.3.2.1

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

$x - 3 = 0$

Step 2.3.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.4

Set $2 x - 5$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Step 2.4.2

Solve $2 x - 5 = 0$ for $x$ .

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Step 2.4.2.1

Add $5$ to both sides of the equation.

$2 x = 5$

Step 2.4.2.2

Divide each term in $2 x = 5$ by $2$ and simplify.

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Step 2.4.2.2.1

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 2.5

Set $x^{2} + 2 x + 5$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} + 2 x + 5$ equal to $0$ .

$x^{2} + 2 x + 5 = 0$

Step 2.5.2

Solve $x^{2} + 2 x + 5 = 0$ 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 $a = 1$ , $b = 2$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

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 $2$ to the power of $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 5}}{2 \cdot 1}$

Step 2.5.2.3.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{- 2 \pm \sqrt{4 - 20}}{2 \cdot 1}$

Step 2.5.2.3.1.3

Subtract $20$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 16}}{2 \cdot 1}$

Step 2.5.2.3.1.4

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

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Step 2.5.2.3.1.5

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

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Step 2.5.2.3.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Step 2.5.2.3.1.7

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

$x = \frac{- 2 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Step 2.5.2.3.1.8

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

$x = \frac{- 2 \pm i \cdot 4}{2 \cdot 1}$

Step 2.5.2.3.1.9

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

$x = \frac{- 2 \pm 4 i}{2 \cdot 1}$

Step 2.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 4 i}{2}$

Step 2.5.2.3.3

Simplify $\frac{- 2 \pm 4 i}{2}$ .

$x = - 1 \pm 2 i$

Step 2.5.2.4

The final answer is the combination of both solutions.

$x = - 1 + 2 i, - 1 - 2 i$

Step 2.6

The final solution is all the values that make ${(x - 3)}^{2} (2 x - 5) (x^{2} + 2 x + 5) = 0$ true.

$x = 3, \frac{5}{2}, - 1 + 2 i, - 1 - 2 i$

Step 3