Precalculus Examples

x^{3} - 4 x^{2} - 7 x + 10

$x^{3} - 4 x^{2} - 7 x + 10$

Step 1

If a polynomial function has integer coefficients, then every rational zero will have the form

\frac{p}{q}

$\frac{p}{q}$ where

p

$p$ is a factor of the constant and

q

$q$ is a factor of the leading coefficient.

p = \pm 1, \pm 2, \pm 5, \pm 10

$p = \pm 1, \pm 2, \pm 5, \pm 10$

q = \pm 1

$q = \pm 1$

Step 2

Find every combination of

\pm \frac{p}{q}

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

\pm 1, \pm 2, \pm 5, \pm 10

$\pm 1, \pm 2, \pm 5, \pm 10$

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

$0$ , which means it is a root.

{(1)}^{3} - 4 {(1)}^{2} - 7 \cdot 1 + 10

${(1)}^{3} - 4 {(1)}^{2} - 7 \cdot 1 + 10$

Step 4

Simplify the expression. In this case, the expression is equal to

0

$0$ so

x = 1

$x = 1$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

One to any power is one.

1 - 4 {(1)}^{2} - 7 \cdot 1 + 10

$1 - 4 {(1)}^{2} - 7 \cdot 1 + 10$

Step 4.1.2

One to any power is one.

1 - 4 \cdot 1 - 7 \cdot 1 + 10

$1 - 4 \cdot 1 - 7 \cdot 1 + 10$

Step 4.1.3

Multiply

- 4

$- 4$ by

1

$1$ .

1 - 4 - 7 \cdot 1 + 10

$1 - 4 - 7 \cdot 1 + 10$

Step 4.1.4

Multiply

- 7

$- 7$ by

1

$1$ .

1 - 4 - 7 + 10

$1 - 4 - 7 + 10$

1 - 4 - 7 + 10

$1 - 4 - 7 + 10$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract

4

$4$ from

1

$1$ .

- 3 - 7 + 10

$- 3 - 7 + 10$

Step 4.2.2

Subtract

7

$7$ from

- 3

$- 3$ .

- 10 + 10

$- 10 + 10$

Step 4.2.3

Add

- 10

$- 10$ and

10

$10$ .

0

$0$

0

$0$

0

$0$

Step 5

Since

1

$1$ is a known root, divide the polynomial by

x - 1

$x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

\frac{x^{3} - 4 x^{2} - 7 x + 10}{x - 1}

$\frac{x^{3} - 4 x^{2} - 7 x + 10}{x - 1}$

Step 6

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

1

$1$ .

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Step 6.1

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

$1$ $1$	$1$ $1$	$- 4$ $- 4$	$- 7$ $- 7$	$10$ $10$

Step 6.2

The first number in the dividend

(1)

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

$1$ $1$	$1$ $1$	$- 4$ $- 4$	$- 7$ $- 7$	$10$ $10$

	$1$ $1$

Step 6.3

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(1)

$(1)$ and place the result of

(1)

$(1)$ under the next term in the dividend

(- 4)

$(- 4)$ .

$1$ $1$	$1$ $1$	$- 4$ $- 4$	$- 7$ $- 7$	$10$ $10$
		$1$ $1$
	$1$ $1$

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.

$1$ $1$	$1$ $1$	$- 4$ $- 4$	$- 7$ $- 7$	$10$ $10$
		$1$ $1$
	$1$ $1$	$- 3$ $- 3$

Step 6.5

Multiply the newest entry in the result

(- 3)

$(- 3)$ by the divisor

(1)

$(1)$ and place the result of

(- 3)

$(- 3)$ under the next term in the dividend

(- 7)

$(- 7)$ .

$1$ $1$	$1$ $1$	$- 4$ $- 4$	$- 7$ $- 7$	$10$ $10$
		$1$ $1$	$- 3$ $- 3$
	$1$ $1$	$- 3$ $- 3$

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.

$1$ $1$	$1$ $1$	$- 4$ $- 4$	$- 7$ $- 7$	$10$ $10$
		$1$ $1$	$- 3$ $- 3$
	$1$ $1$	$- 3$ $- 3$	$- 10$ $- 10$

Step 6.7

Multiply the newest entry in the result

(- 10)

$(- 10)$ by the divisor

(1)

$(1)$ and place the result of

(- 10)

$(- 10)$ under the next term in the dividend

(10)

$(10)$ .

$1$ $1$	$1$ $1$	$- 4$ $- 4$	$- 7$ $- 7$	$10$ $10$
		$1$ $1$	$- 3$ $- 3$	$- 10$ $- 10$
	$1$ $1$	$- 3$ $- 3$	$- 10$ $- 10$

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.

$1$ $1$	$1$ $1$	$- 4$ $- 4$	$- 7$ $- 7$	$10$ $10$
		$1$ $1$	$- 3$ $- 3$	$- 10$ $- 10$
	$1$ $1$	$- 3$ $- 3$	$- 10$ $- 10$	$0$ $0$

Step 6.9

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

1 x^{2} + - 3 x - 10

$1 x^{2} + - 3 x - 10$

Step 6.10

Simplify the quotient polynomial.

x^{2} - 3 x - 10

$x^{2} - 3 x - 10$

x^{2} - 3 x - 10

$x^{2} - 3 x - 10$

Step 7

Factor

x^{2} - 3 x - 10

$x^{2} - 3 x - 10$ using the AC method.

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Step 7.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

- 10

$- 10$ and whose sum is

- 3

$- 3$ .

- 5, 2

$- 5, 2$

Step 7.2

Write the factored form using these integers.

(x - 1) + (x - 5) (x + 2)

$(x - 1) + (x - 5) (x + 2)$

(x - 1) (x - 5) (x + 2)

$(x - 1) (x - 5) (x + 2)$

Step 8

Factor the left side of the equation.

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Step 8.1

Factor

x^{3} - 4 x^{2} - 7 x + 10

$x^{3} - 4 x^{2} - 7 x + 10$ using the rational roots test.

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Step 8.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form

\frac{p}{q}

$\frac{p}{q}$ where

p

$p$ is a factor of the constant and

q

$q$ is a factor of the leading coefficient.

p = \pm 1, \pm 10, \pm 2, \pm 5

$p = \pm 1, \pm 10, \pm 2, \pm 5$

$q = \pm 1$

Step 8.1.2

Find every combination of

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

$\pm 1, \pm 10, \pm 2, \pm 5$

Step 8.1.3

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 8.1.3.1

Substitute

$1$ into the polynomial.

$1^{3} - 4 \cdot 1^{2} - 7 \cdot 1 + 10$

Step 8.1.3.2

Raise

$1$ to the power of

$3$ .

$1 - 4 \cdot 1^{2} - 7 \cdot 1 + 10$

Step 8.1.3.3

Raise

$1$ to the power of

$2$ .

$1 - 4 \cdot 1 - 7 \cdot 1 + 10$

Step 8.1.3.4

Multiply

$- 4$ by

$1$ .

$1 - 4 - 7 \cdot 1 + 10$

Step 8.1.3.5

Subtract

$4$ from

$1$ .

$- 3 - 7 \cdot 1 + 10$

Step 8.1.3.6

Multiply

$- 7$ by

$1$ .

$- 3 - 7 + 10$

Step 8.1.3.7

Subtract

$7$ from

$- 3$ .

$- 10 + 10$

Step 8.1.3.8

Add

$- 10$ and

$10$ .

$0$

Step 8.1.4

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.

$\frac{x^{3} - 4 x^{2} - 7 x + 10}{x - 1}$

Step 8.1.5

Divide

$x^{3} - 4 x^{2} - 7 x + 10$ by

$x - 1$ .

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Step 8.1.5.1

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

$0$ .

$x$

$1$

$x^{3}$

$4 x^{2}$

$7 x$

$10$

Step 8.1.5.2

Divide the highest order term in the dividend

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

$x$ .

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$

Step 8.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			+	$x^{3}$	-	$x^{2}$

Step 8.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$

Step 8.1.5.5

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$

Step 8.1.5.6

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$7 x$

Step 8.1.5.7

Divide the highest order term in the dividend

$- 3 x^{2}$ by the highest order term in divisor

$x$ .

				$x^{2}$	-	$3 x$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$7 x$

Step 8.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	+	$3 x$

Step 8.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in

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

				$x^{2}$	-	$3 x$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$7 x$
					+	$3 x^{2}$	-	$3 x$

Step 8.1.5.10

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

				$x^{2}$	-	$3 x$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$7 x$
					+	$3 x^{2}$	-	$3 x$
							-	$10 x$

Step 8.1.5.11

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

				$x^{2}$	-	$3 x$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$7 x$
					+	$3 x^{2}$	-	$3 x$
							-	$10 x$	+	$10$

Step 8.1.5.12

Divide the highest order term in the dividend

$- 10 x$ by the highest order term in divisor

$x$ .

				$x^{2}$	-	$3 x$	-	$10$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$7 x$
					+	$3 x^{2}$	-	$3 x$
							-	$10 x$	+	$10$

Step 8.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$	-	$10$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$7 x$
					+	$3 x^{2}$	-	$3 x$
							-	$10 x$	+	$10$
							-	$10 x$	+	$10$

Step 8.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in

$- 10 x + 10$

				$x^{2}$	-	$3 x$	-	$10$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$7 x$
					+	$3 x^{2}$	-	$3 x$
							-	$10 x$	+	$10$
							+	$10 x$	-	$10$

Step 8.1.5.15

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

				$x^{2}$	-	$3 x$	-	$10$
$x$	-	$1$		$x^{3}$	-	$4 x^{2}$	-	$7 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$3 x^{2}$	-	$7 x$
					+	$3 x^{2}$	-	$3 x$
							-	$10 x$	+	$10$
							+	$10 x$	-	$10$
										$0$

Step 8.1.5.16

Since the remander is

$0$ , the final answer is the quotient.

$x^{2} - 3 x - 10$

Step 8.1.6

Write

$x^{3} - 4 x^{2} - 7 x + 10$ as a set of factors.

$(x - 1) (x^{2} - 3 x - 10) = 0$

Step 8.2

Factor

$x^{2} - 3 x - 10$ using the AC method.

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Step 8.2.1

Factor

$x^{2} - 3 x - 10$ using the AC method.

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Step 8.2.1.1

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

$- 10$ and whose sum is

$- 3$ .

$- 5, 2$

Step 8.2.1.2

Write the factored form using these integers.

$(x - 1) ((x - 5) (x + 2)) = 0$

Step 8.2.2

Remove unnecessary parentheses.

$(x - 1) (x - 5) (x + 2) = 0$

Step 9

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$x - 1 = 0$

$x - 5 = 0$

$x + 2 = 0$

Step 10

Set

$x - 1$ equal to

$0$ and solve for

$x$ .

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Step 10.1

Set

$x - 1$ equal to

$0$ .

$x - 1 = 0$

Step 10.2

Add

$1$ to both sides of the equation.

$x = 1$

Step 11

Set

$x - 5$ equal to

$0$ and solve for

$x$ .

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Step 11.1

Set

$x - 5$ equal to

$0$ .

$x - 5 = 0$

Step 11.2

Add

$5$ to both sides of the equation.

$x = 5$

Step 12

Set

$x + 2$ equal to

$0$ and solve for

$x$ .

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Step 12.1

Set

$x + 2$ equal to

$0$ .

$x + 2 = 0$

Step 12.2

Subtract

$2$ from both sides of the equation.

$x = - 2$

Step 13

The final solution is all the values that make

$(x - 1) (x - 5) (x + 2) = 0$ true.

$x = 1, 5, - 2$

Step 14