Calculus Examples

$f (x) = x^{3} + 4 x^{2} + 8 x + 5$

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.

$p = \pm 1, \pm 5$

$q = \pm 1$

Step 1.2

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

$\pm 1, \pm 5$

Step 2

Apply synthetic division on $\frac{x^{3} + 4 x^{2} + 8 x + 5}{x - 1}$ when $x = 1$ .

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Step 2.1

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

$1$	$1$	$4$	$8$	$5$

Step 2.2

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

$1$	$1$	$4$	$8$	$5$

	$1$

Step 2.3

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 $(4)$ .

$1$	$1$	$4$	$8$	$5$
		$1$
	$1$

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.

$1$	$1$	$4$	$8$	$5$
		$1$
	$1$	$5$

Step 2.5

Multiply the newest entry in the result $(5)$ by the divisor $(1)$ and place the result of $(5)$ under the next term in the dividend $(8)$ .

$1$	$1$	$4$	$8$	$5$
		$1$	$5$
	$1$	$5$

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.

$1$	$1$	$4$	$8$	$5$
		$1$	$5$
	$1$	$5$	$13$

Step 2.7

Multiply the newest entry in the result $(13)$ by the divisor $(1)$ and place the result of $(13)$ under the next term in the dividend $(5)$ .

$1$	$1$	$4$	$8$	$5$
		$1$	$5$	$13$
	$1$	$5$	$13$

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.

$1$	$1$	$4$	$8$	$5$
		$1$	$5$	$13$
	$1$	$5$	$13$	$18$

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.

$1 x^{2} + 5 x + 13 + \frac{18}{x - 1}$

Step 2.10

Simplify the quotient polynomial.

$x^{2} + 5 x + 13 + \frac{18}{x - 1}$

Step 3

Since $1 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $1$ is an upper bound for the real roots of the function.

Upper Bound: $1$

Step 4

Apply synthetic division on $\frac{x^{3} + 4 x^{2} + 8 x + 5}{x - 5}$ when $x = 5$ .

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Step 4.1

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

$5$	$1$	$4$	$8$	$5$

Step 4.2

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

$5$	$1$	$4$	$8$	$5$

	$1$

Step 4.3

Multiply the newest entry in the result $(1)$ by the divisor $(5)$ and place the result of $(5)$ under the next term in the dividend $(4)$ .

$5$	$1$	$4$	$8$	$5$
		$5$
	$1$

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.

$5$	$1$	$4$	$8$	$5$
		$5$
	$1$	$9$

Step 4.5

Multiply the newest entry in the result $(9)$ by the divisor $(5)$ and place the result of $(45)$ under the next term in the dividend $(8)$ .

$5$	$1$	$4$	$8$	$5$
		$5$	$45$
	$1$	$9$

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.

$5$	$1$	$4$	$8$	$5$
		$5$	$45$
	$1$	$9$	$53$

Step 4.7

Multiply the newest entry in the result $(53)$ by the divisor $(5)$ and place the result of $(265)$ under the next term in the dividend $(5)$ .

$5$	$1$	$4$	$8$	$5$
		$5$	$45$	$265$
	$1$	$9$	$53$

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.

$5$	$1$	$4$	$8$	$5$
		$5$	$45$	$265$
	$1$	$9$	$53$	$270$

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.

$1 x^{2} + 9 x + 53 + \frac{270}{x - 5}$

Step 4.10

Simplify the quotient polynomial.

$x^{2} + 9 x + 53 + \frac{270}{x - 5}$

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 $\frac{x^{3} + 4 x^{2} + 8 x + 5}{x + 5}$ when $x = - 5$ .

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Step 6.1

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

$- 5$	$1$	$4$	$8$	$5$

Step 6.2

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

$- 5$	$1$	$4$	$8$	$5$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(- 5)$ and place the result of $(- 5)$ under the next term in the dividend $(4)$ .

$- 5$	$1$	$4$	$8$	$5$
		$- 5$
	$1$

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.

$- 5$	$1$	$4$	$8$	$5$
		$- 5$
	$1$	$- 1$

Step 6.5

Multiply the newest entry in the result $(- 1)$ by the divisor $(- 5)$ and place the result of $(5)$ under the next term in the dividend $(8)$ .

$- 5$	$1$	$4$	$8$	$5$
		$- 5$	$5$
	$1$	$- 1$

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.

$- 5$	$1$	$4$	$8$	$5$
		$- 5$	$5$
	$1$	$- 1$	$13$

Step 6.7

Multiply the newest entry in the result $(13)$ by the divisor $(- 5)$ and place the result of $(- 65)$ under the next term in the dividend $(5)$ .

$- 5$	$1$	$4$	$8$	$5$
		$- 5$	$5$	$- 65$
	$1$	$- 1$	$13$

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.

$- 5$	$1$	$4$	$8$	$5$
		$- 5$	$5$	$- 65$
	$1$	$- 1$	$13$	$- 60$

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.

$1 x^{2} + - 1 x + 13 + \frac{- 60}{x + 5}$

Step 6.10

Simplify the quotient polynomial.

$x^{2} - x + 13 - \frac{60}{x + 5}$

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

Determine the upper and lower bounds.

Upper Bounds: $1, 5$

Lower Bound: $- 5$

Step 9