Precalculus Examples

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

Step 1

Find every combination of $\pm \frac{p}{q}$ .

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Step 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.

$p = \pm 1, \pm 5$

$q = \pm 1, \pm 2$

Step 1.2

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

$\pm 1, \pm \frac{1}{2}, \pm 5, \pm \frac{5}{2}$

Step 2

Apply synthetic division on $\frac{2 x^{3} - 2 x^{2} - 3 x - 5}{x + 1}$ when $x = - 1$ .

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Step 2.1

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

$- 1$	$2$	$- 2$	$- 3$	$- 5$

Step 2.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$2$	$- 2$	$- 3$	$- 5$

	$2$

Step 2.3

Multiply the newest entry in the result $(2)$ by the divisor $(- 1)$ and place the result of $(- 2)$ under the next term in the dividend $(- 2)$ .

$- 1$	$2$	$- 2$	$- 3$	$- 5$
		$- 2$
	$2$

Step 2.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$2$	$- 2$	$- 3$	$- 5$
		$- 2$
	$2$	$- 4$

Step 2.5

Multiply the newest entry in the result $(- 4)$ by the divisor $(- 1)$ and place the result of $(4)$ under the next term in the dividend $(- 3)$ .

$- 1$	$2$	$- 2$	$- 3$	$- 5$
		$- 2$	$4$
	$2$	$- 4$

Step 2.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$2$	$- 2$	$- 3$	$- 5$
		$- 2$	$4$
	$2$	$- 4$	$1$

Step 2.7

Multiply the newest entry in the result $(1)$ by the divisor $(- 1)$ and place the result of $(- 1)$ under the next term in the dividend $(- 5)$ .

$- 1$	$2$	$- 2$	$- 3$	$- 5$
		$- 2$	$4$	$- 1$
	$2$	$- 4$	$1$

Step 2.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$2$	$- 2$	$- 3$	$- 5$
		$- 2$	$4$	$- 1$
	$2$	$- 4$	$1$	$- 6$

Step 2.9

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

$2 x^{2} + - 4 x + 1 + \frac{- 6}{x + 1}$

Step 2.10

Simplify the quotient polynomial.

$2 x^{2} - 4 x + 1 - \frac{6}{x + 1}$

Step 3

Since $- 1 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 1$ is a lower bound for the real roots of the function.

Lower Bound: $- 1$

Step 4

Apply synthetic division on $\frac{2 x^{3} - 2 x^{2} - 3 x - 5}{x - 5}$ when $x = 5$ .

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Step 4.1

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

$5$	$2$	$- 2$	$- 3$	$- 5$

Step 4.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$5$	$2$	$- 2$	$- 3$	$- 5$

	$2$

Step 4.3

Multiply the newest entry in the result $(2)$ by the divisor $(5)$ and place the result of $(10)$ under the next term in the dividend $(- 2)$ .

$5$	$2$	$- 2$	$- 3$	$- 5$
		$10$
	$2$

Step 4.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$5$	$2$	$- 2$	$- 3$	$- 5$
		$10$
	$2$	$8$

Step 4.5

Multiply the newest entry in the result $(8)$ by the divisor $(5)$ and place the result of $(40)$ under the next term in the dividend $(- 3)$ .

$5$	$2$	$- 2$	$- 3$	$- 5$
		$10$	$40$
	$2$	$8$

Step 4.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$5$	$2$	$- 2$	$- 3$	$- 5$
		$10$	$40$
	$2$	$8$	$37$

Step 4.7

Multiply the newest entry in the result $(37)$ by the divisor $(5)$ and place the result of $(185)$ under the next term in the dividend $(- 5)$ .

$5$	$2$	$- 2$	$- 3$	$- 5$
		$10$	$40$	$185$
	$2$	$8$	$37$

Step 4.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$5$	$2$	$- 2$	$- 3$	$- 5$
		$10$	$40$	$185$
	$2$	$8$	$37$	$180$

Step 4.9

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

$2 x^{2} + 8 x + 37 + \frac{180}{x - 5}$

Step 5

Since $5 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $5$ is an upper bound for the real roots of the function.

Upper Bound: $5$

Step 6

Apply synthetic division on $\frac{2 x^{3} - 2 x^{2} - 3 x - 5}{x + 5}$ when $x = - 5$ .

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Step 6.1

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

$- 5$	$2$	$- 2$	$- 3$	$- 5$

Step 6.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- 5$	$2$	$- 2$	$- 3$	$- 5$

	$2$

Step 6.3

Multiply the newest entry in the result $(2)$ by the divisor $(- 5)$ and place the result of $(- 10)$ under the next term in the dividend $(- 2)$ .

$- 5$	$2$	$- 2$	$- 3$	$- 5$
		$- 10$
	$2$

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.

$- 5$	$2$	$- 2$	$- 3$	$- 5$
		$- 10$
	$2$	$- 12$

Step 6.5

Multiply the newest entry in the result $(- 12)$ by the divisor $(- 5)$ and place the result of $(60)$ under the next term in the dividend $(- 3)$ .

$- 5$	$2$	$- 2$	$- 3$	$- 5$
		$- 10$	$60$
	$2$	$- 12$

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.

$- 5$	$2$	$- 2$	$- 3$	$- 5$
		$- 10$	$60$
	$2$	$- 12$	$57$

Step 6.7

Multiply the newest entry in the result $(57)$ by the divisor $(- 5)$ and place the result of $(- 285)$ under the next term in the dividend $(- 5)$ .

$- 5$	$2$	$- 2$	$- 3$	$- 5$
		$- 10$	$60$	$- 285$
	$2$	$- 12$	$57$

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.

$- 5$	$2$	$- 2$	$- 3$	$- 5$
		$- 10$	$60$	$- 285$
	$2$	$- 12$	$57$	$- 290$

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.

$2 x^{2} + - 12 x + 57 + \frac{- 290}{x + 5}$

Step 6.10

Simplify the quotient polynomial.

$2 x^{2} - 12 x + 57 - \frac{290}{x + 5}$

Step 7

Since $- 5 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 5$ is a lower bound for the real roots of the function.

Lower Bound: $- 5$

Step 8

Apply synthetic division on $\frac{2 x^{3} - 2 x^{2} - 3 x - 5}{x - \frac{5}{2}}$ when $x = \frac{5}{2}$ .

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Step 8.1

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

$\frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$

Step 8.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$\frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$

	$2$

Step 8.3

Multiply the newest entry in the result $(2)$ by the divisor $(\frac{5}{2})$ and place the result of $(5)$ under the next term in the dividend $(- 2)$ .

$\frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$
		$5$
	$2$

Step 8.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$
		$5$
	$2$	$3$

Step 8.5

Multiply the newest entry in the result $(3)$ by the divisor $(\frac{5}{2})$ and place the result of $(\frac{15}{2})$ under the next term in the dividend $(- 3)$ .

$\frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$
		$5$	$\frac{15}{2}$
	$2$	$3$

Step 8.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$
		$5$	$\frac{15}{2}$
	$2$	$3$	$\frac{9}{2}$

Step 8.7

Multiply the newest entry in the result $(\frac{9}{2})$ by the divisor $(\frac{5}{2})$ and place the result of $(\frac{45}{4})$ under the next term in the dividend $(- 5)$ .

$\frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$
		$5$	$\frac{15}{2}$	$\frac{45}{4}$
	$2$	$3$	$\frac{9}{2}$

Step 8.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$
		$5$	$\frac{15}{2}$	$\frac{45}{4}$
	$2$	$3$	$\frac{9}{2}$	$\frac{25}{4}$

Step 8.9

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

$2 x^{2} + 3 x + \frac{9}{2} + \frac{\frac{25}{4}}{x - \frac{5}{2}}$

Step 8.10

Simplify the quotient polynomial.

$2 x^{2} + 3 x + \frac{9}{2} + \frac{25}{2 (2 x - 5)}$

Step 9

Since $\frac{5}{2} > 0$ and all of the signs in the bottom row of the synthetic division are positive, $\frac{5}{2}$ is an upper bound for the real roots of the function.

Upper Bound: $\frac{5}{2}$

Step 10

Apply synthetic division on $\frac{2 x^{3} - 2 x^{2} - 3 x - 5}{x + \frac{5}{2}}$ when $x = - \frac{5}{2}$ .

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Step 10.1

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

$- \frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$

Step 10.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- \frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$

	$2$

Step 10.3

Multiply the newest entry in the result $(2)$ by the divisor $(- \frac{5}{2})$ and place the result of $(- 5)$ under the next term in the dividend $(- 2)$ .

$- \frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$
		$- 5$
	$2$

Step 10.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$
		$- 5$
	$2$	$- 7$

Step 10.5

Multiply the newest entry in the result $(- 7)$ by the divisor $(- \frac{5}{2})$ and place the result of $(\frac{35}{2})$ under the next term in the dividend $(- 3)$ .

$- \frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$
		$- 5$	$\frac{35}{2}$
	$2$	$- 7$

Step 10.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$
		$- 5$	$\frac{35}{2}$
	$2$	$- 7$	$\frac{29}{2}$

Step 10.7

Multiply the newest entry in the result $(\frac{29}{2})$ by the divisor $(- \frac{5}{2})$ and place the result of $(- \frac{145}{4})$ under the next term in the dividend $(- 5)$ .

$- \frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$
		$- 5$	$\frac{35}{2}$	$- \frac{145}{4}$
	$2$	$- 7$	$\frac{29}{2}$

Step 10.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{5}{2}$	$2$	$- 2$	$- 3$	$- 5$
		$- 5$	$\frac{35}{2}$	$- \frac{145}{4}$
	$2$	$- 7$	$\frac{29}{2}$	$- \frac{165}{4}$

Step 10.9

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

$2 x^{2} + - 7 x + \frac{29}{2} + \frac{- \frac{165}{4}}{x + \frac{5}{2}}$

Step 10.10

Simplify the quotient polynomial.

$2 x^{2} - 7 x + \frac{29}{2} - \frac{165}{2 (2 x + 5)}$

Step 11

Since $- \frac{5}{2} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{5}{2}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{5}{2}$

Step 12

Determine the upper and lower bounds.

Upper Bounds: $5, \frac{5}{2}$

Lower Bounds: $- 1, - 5, - \frac{5}{2}$

Step 13