Algebra Examples

y = x^{3} - 4 x^{2} - 11 x + 30

$y = x^{3} - 4 x^{2} - 11 x + 30$

Step 1

Set

x^{3} - 4 x^{2} - 11 x + 30

$x^{3} - 4 x^{2} - 11 x + 30$ equal to

0

$0$ .

x^{3} - 4 x^{2} - 11 x + 30 = 0

$x^{3} - 4 x^{2} - 11 x + 30 = 0$

Step 2

Solve for

x

$x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor

x^{3} - 4 x^{2} - 11 x + 30

$x^{3} - 4 x^{2} - 11 x + 30$ 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}

$\frac{p}{q}$ where

p

$p$ is a factor of the constant and

q

$q$ is a factor of the leading coefficient.

p = \pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6

$p = \pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6$

q = \pm 1

$q = \pm 1$

Step 2.1.1.2

Find every combination of

\pm \frac{p}{q}

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

\pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6

$\pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6$

Step 2.1.1.3

Substitute

2

$2$ and simplify the expression. In this case, the expression is equal to

0

$0$ so

2

$2$ is a root of the polynomial.

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Step 2.1.1.3.1

Substitute

2

$2$ into the polynomial.

2^{3} - 4 \cdot 2^{2} - 11 \cdot 2 + 30

$2^{3} - 4 \cdot 2^{2} - 11 \cdot 2 + 30$

Step 2.1.1.3.2

Raise

2

$2$ to the power of

3

$3$ .

8 - 4 \cdot 2^{2} - 11 \cdot 2 + 30

$8 - 4 \cdot 2^{2} - 11 \cdot 2 + 30$

Step 2.1.1.3.3

Raise

2

$2$ to the power of

2

$2$ .

8 - 4 \cdot 4 - 11 \cdot 2 + 30

$8 - 4 \cdot 4 - 11 \cdot 2 + 30$

Step 2.1.1.3.4

Multiply

- 4

$- 4$ by

4

$4$ .

8 - 16 - 11 \cdot 2 + 30

$8 - 16 - 11 \cdot 2 + 30$

Step 2.1.1.3.5

Subtract

16

$16$ from

8

$8$ .

- 8 - 11 \cdot 2 + 30

$- 8 - 11 \cdot 2 + 30$

Step 2.1.1.3.6

Multiply

- 11

$- 11$ by

2

$2$ .

- 8 - 22 + 30

$- 8 - 22 + 30$

Step 2.1.1.3.7

Subtract

22

$22$ from

- 8

$- 8$ .

- 30 + 30

$- 30 + 30$

Step 2.1.1.3.8

Add

- 30

$- 30$ and

30

$30$ .

0

$0$

0

$0$

Step 2.1.1.4

Since

2

$2$ is a known root, divide the polynomial by

x - 2

$x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

\frac{x^{3} - 4 x^{2} - 11 x + 30}{x - 2}

$\frac{x^{3} - 4 x^{2} - 11 x + 30}{x - 2}$

Step 2.1.1.5

Divide

x^{3} - 4 x^{2} - 11 x + 30

$x^{3} - 4 x^{2} - 11 x + 30$ by

x - 2

$x - 2$ .

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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

$0$ .

x

$x$

2

$2$

x^{3}

$x^{3}$

4 x^{2}

$4 x^{2}$

11 x

$11 x$

30

$30$

Step 2.1.1.5.2

Divide the highest order term in the dividend

x^{3}

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

x

$x$ .

					$x^{2}$ $x^{2}$
	$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$ $x^{2}$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			+	$x^{3}$ $x^{3}$	-	$2 x^{2}$ $2 x^{2}$

Step 2.1.1.5.4

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

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

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

				$x^{2}$ $x^{2}$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			-	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$

Step 2.1.1.5.5

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

				$x^{2}$ $x^{2}$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			-	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$

Step 2.1.1.5.6

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

				$x^{2}$ $x^{2}$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			-	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$11 x$ $11 x$

Step 2.1.1.5.7

Divide the highest order term in the dividend

- 2 x^{2}

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

x

$x$ .

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			-	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$11 x$ $11 x$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			-	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$11 x$ $11 x$
					-	$2 x^{2}$ $2 x^{2}$	+	$4 x$ $4 x$

Step 2.1.1.5.9

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

- 2 x^{2} + 4 x

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

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			-	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$11 x$ $11 x$
					+	$2 x^{2}$ $2 x^{2}$	-	$4 x$ $4 x$

Step 2.1.1.5.10

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

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			-	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$11 x$ $11 x$
					+	$2 x^{2}$ $2 x^{2}$	-	$4 x$ $4 x$
							-	$15 x$ $15 x$

Step 2.1.1.5.11

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

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			-	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$11 x$ $11 x$
					+	$2 x^{2}$ $2 x^{2}$	-	$4 x$ $4 x$
							-	$15 x$ $15 x$	+	$30$ $30$

Step 2.1.1.5.12

Divide the highest order term in the dividend

- 15 x

$- 15 x$ by the highest order term in divisor

x

$x$ .

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$	-	$15$ $15$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			-	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$11 x$ $11 x$
					+	$2 x^{2}$ $2 x^{2}$	-	$4 x$ $4 x$
							-	$15 x$ $15 x$	+	$30$ $30$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$	-	$15$ $15$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			-	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$11 x$ $11 x$
					+	$2 x^{2}$ $2 x^{2}$	-	$4 x$ $4 x$
							-	$15 x$ $15 x$	+	$30$ $30$
							-	$15 x$ $15 x$	+	$30$ $30$

Step 2.1.1.5.14

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

- 15 x + 30

$- 15 x + 30$

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$	-	$15$ $15$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			-	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$11 x$ $11 x$
					+	$2 x^{2}$ $2 x^{2}$	-	$4 x$ $4 x$
							-	$15 x$ $15 x$	+	$30$ $30$
							+	$15 x$ $15 x$	-	$30$ $30$

Step 2.1.1.5.15

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

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$	-	$15$ $15$
$x$ $x$	-	$2$ $2$		$x^{3}$ $x^{3}$	-	$4 x^{2}$ $4 x^{2}$	-	$11 x$ $11 x$	+	$30$ $30$
			-	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$11 x$ $11 x$
					+	$2 x^{2}$ $2 x^{2}$	-	$4 x$ $4 x$
							-	$15 x$ $15 x$	+	$30$ $30$
							+	$15 x$ $15 x$	-	$30$ $30$
										$0$ $0$

Step 2.1.1.5.16

Since the remander is

0

$0$ , the final answer is the quotient.

x^{2} - 2 x - 15

$x^{2} - 2 x - 15$

x^{2} - 2 x - 15

$x^{2} - 2 x - 15$

Step 2.1.1.6

Write

x^{3} - 4 x^{2} - 11 x + 30

$x^{3} - 4 x^{2} - 11 x + 30$ as a set of factors.

(x - 2) (x^{2} - 2 x - 15) = 0

$(x - 2) (x^{2} - 2 x - 15) = 0$

(x - 2) (x^{2} - 2 x - 15) = 0

$(x - 2) (x^{2} - 2 x - 15) = 0$

Step 2.1.2

Factor

x^{2} - 2 x - 15

$x^{2} - 2 x - 15$ using the AC method.

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Step 2.1.2.1

Factor

x^{2} - 2 x - 15

$x^{2} - 2 x - 15$ using the AC method.

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Step 2.1.2.1.1

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

- 15

$- 15$ and whose sum is

- 2

$- 2$ .

- 5, 3

$- 5, 3$

Step 2.1.2.1.2

Write the factored form using these integers.

(x - 2) ((x - 5) (x + 3)) = 0

$(x - 2) ((x - 5) (x + 3)) = 0$

(x - 2) ((x - 5) (x + 3)) = 0

$(x - 2) ((x - 5) (x + 3)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

(x - 2) (x - 5) (x + 3) = 0

$(x - 2) (x - 5) (x + 3) = 0$

(x - 2) (x - 5) (x + 3) = 0

$(x - 2) (x - 5) (x + 3) = 0$

(x - 2) (x - 5) (x + 3) = 0

$(x - 2) (x - 5) (x + 3) = 0$

Step 2.2

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

0

$0$ , the entire expression will be equal to

0

$0$ .

x - 2 = 0

$x - 2 = 0$

x - 5 = 0

$x - 5 = 0$

x + 3 = 0

$x + 3 = 0$

Step 2.3

Set

x - 2

$x - 2$ equal to

0

$0$ and solve for

x

$x$ .

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Step 2.3.1

Set

x - 2

$x - 2$ equal to

0

$0$ .

x - 2 = 0

$x - 2 = 0$

Step 2.3.2

Add

2

$2$ to both sides of the equation.

x = 2

$x = 2$

x = 2

$x = 2$

Step 2.4

Set

x - 5

$x - 5$ equal to

0

$0$ and solve for

x

$x$ .

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Step 2.4.1

Set

x - 5

$x - 5$ equal to

0

$0$ .

x - 5 = 0

$x - 5 = 0$

Step 2.4.2

Add

5

$5$ to both sides of the equation.

x = 5

$x = 5$

x = 5

$x = 5$

Step 2.5

Set

x + 3

$x + 3$ equal to

0

$0$ and solve for

x

$x$ .

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Step 2.5.1

Set

x + 3

$x + 3$ equal to

0

$0$ .

x + 3 = 0

$x + 3 = 0$

Step 2.5.2

Subtract

3

$3$ from both sides of the equation.

x = - 3

$x = - 3$

x = - 3

$x = - 3$

Step 2.6

The final solution is all the values that make

(x - 2) (x - 5) (x + 3) = 0

$(x - 2) (x - 5) (x + 3) = 0$ true. The multiplicity of a root is the number of times the root appears.

x = 2

$x = 2$ (Multiplicity of

1

$1$ )

x = 5

$x = 5$ (Multiplicity of

1

$1$ )

x = - 3

$x = - 3$ (Multiplicity of

1

$1$ )

x = 2

$x = 2$ (Multiplicity of

1

$1$ )

x = 5

$x = 5$ (Multiplicity of

1

$1$ )

x = - 3

$x = - 3$ (Multiplicity of

1

$1$ )

Step 3

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