Calculus Examples

$F (x) = 2 x^{3} + 3 x^{2} - 13 x - 15$

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.

$p = \pm 1, \pm 3, \pm 5, \pm 15$

$q = \pm 1, \pm 2$

Step 1.2

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

$\pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 5, \pm \frac{5}{2}, \pm 15, \pm \frac{15}{2}$

Step 2

Apply synthetic division on $\frac{2 x^{3} + 3 x^{2} - 13 x - 15}{x - 3}$ when $x = 3$ .

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Step 2.1

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

$3$	$2$	$3$	$- 13$	$- 15$

Step 2.2

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

$3$	$2$	$3$	$- 13$	$- 15$

	$2$

Step 2.3

Multiply the newest entry in the result $(2)$ by the divisor $(3)$ and place the result of $(6)$ under the next term in the dividend $(3)$ .

$3$	$2$	$3$	$- 13$	$- 15$
		$6$
	$2$

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.

$3$	$2$	$3$	$- 13$	$- 15$
		$6$
	$2$	$9$

Step 2.5

Multiply the newest entry in the result $(9)$ by the divisor $(3)$ and place the result of $(27)$ under the next term in the dividend $(- 13)$ .

$3$	$2$	$3$	$- 13$	$- 15$
		$6$	$27$
	$2$	$9$

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.

$3$	$2$	$3$	$- 13$	$- 15$
		$6$	$27$
	$2$	$9$	$14$

Step 2.7

Multiply the newest entry in the result $(14)$ by the divisor $(3)$ and place the result of $(42)$ under the next term in the dividend $(- 15)$ .

$3$	$2$	$3$	$- 13$	$- 15$
		$6$	$27$	$42$
	$2$	$9$	$14$

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.

$3$	$2$	$3$	$- 13$	$- 15$
		$6$	$27$	$42$
	$2$	$9$	$14$	$27$

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.

$2 x^{2} + 9 x + 14 + \frac{27}{x - 3}$

Step 3

Since $3 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $3$ is an upper bound for the real roots of the function.

Upper Bound: $3$

Step 4

Apply synthetic division on $\frac{2 x^{3} + 3 x^{2} - 13 x - 15}{x + \frac{3}{2}}$ when $x = - \frac{3}{2}$ .

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Step 4.1

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

$- \frac{3}{2}$	$2$	$3$	$- 13$	$- 15$

Step 4.2

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

$- \frac{3}{2}$	$2$	$3$	$- 13$	$- 15$

	$2$

Step 4.3

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

$- \frac{3}{2}$	$2$	$3$	$- 13$	$- 15$
		$- 3$
	$2$

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.

$- \frac{3}{2}$	$2$	$3$	$- 13$	$- 15$
		$- 3$
	$2$	$0$

Step 4.5

Multiply the newest entry in the result $(0)$ by the divisor $(- \frac{3}{2})$ and place the result of $(0)$ under the next term in the dividend $(- 13)$ .

$- \frac{3}{2}$	$2$	$3$	$- 13$	$- 15$
		$- 3$	$0$
	$2$	$0$

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.

$- \frac{3}{2}$	$2$	$3$	$- 13$	$- 15$
		$- 3$	$0$
	$2$	$0$	$- 13$

Step 4.7

Multiply the newest entry in the result $(- 13)$ by the divisor $(- \frac{3}{2})$ and place the result of $(\frac{39}{2})$ under the next term in the dividend $(- 15)$ .

$- \frac{3}{2}$	$2$	$3$	$- 13$	$- 15$
		$- 3$	$0$	$\frac{39}{2}$
	$2$	$0$	$- 13$

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.

$- \frac{3}{2}$	$2$	$3$	$- 13$	$- 15$
		$- 3$	$0$	$\frac{39}{2}$
	$2$	$0$	$- 13$	$\frac{9}{2}$

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.

$2 x^{2} + 0 x - 13 + \frac{\frac{9}{2}}{x + \frac{3}{2}}$

Step 4.10

Simplify the quotient polynomial.

$2 x^{2} - 13 + \frac{9}{2 x + 3}$

Step 5

Since $- \frac{3}{2} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{3}{2}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{3}{2}$

Step 6

Apply synthetic division on $\frac{2 x^{3} + 3 x^{2} - 13 x - 15}{x - 5}$ when $x = 5$ .

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Step 6.1

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

$5$	$2$	$3$	$- 13$	$- 15$

Step 6.2

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

$5$	$2$	$3$	$- 13$	$- 15$

	$2$

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 $(3)$ .

$5$	$2$	$3$	$- 13$	$- 15$
		$10$
	$2$

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.

$5$	$2$	$3$	$- 13$	$- 15$
		$10$
	$2$	$13$

Step 6.5

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 $(- 13)$ .

$5$	$2$	$3$	$- 13$	$- 15$
		$10$	$65$
	$2$	$13$

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.

$5$	$2$	$3$	$- 13$	$- 15$
		$10$	$65$
	$2$	$13$	$52$

Step 6.7

Multiply the newest entry in the result $(52)$ by the divisor $(5)$ and place the result of $(260)$ under the next term in the dividend $(- 15)$ .

$5$	$2$	$3$	$- 13$	$- 15$
		$10$	$65$	$260$
	$2$	$13$	$52$

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.

$5$	$2$	$3$	$- 13$	$- 15$
		$10$	$65$	$260$
	$2$	$13$	$52$	$245$

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.

$2 x^{2} + 13 x + 52 + \frac{245}{x - 5}$

Step 7

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 8

Apply synthetic division on $\frac{2 x^{3} + 3 x^{2} - 13 x - 15}{x + 5}$ when $x = - 5$ .

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Step 8.1

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

$- 5$	$2$	$3$	$- 13$	$- 15$

Step 8.2

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

$- 5$	$2$	$3$	$- 13$	$- 15$

	$2$

Step 8.3

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 $(3)$ .

$- 5$	$2$	$3$	$- 13$	$- 15$
		$- 10$
	$2$

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.

$- 5$	$2$	$3$	$- 13$	$- 15$
		$- 10$
	$2$	$- 7$

Step 8.5

Multiply the newest entry in the result $(- 7)$ by the divisor $(- 5)$ and place the result of $(35)$ under the next term in the dividend $(- 13)$ .

$- 5$	$2$	$3$	$- 13$	$- 15$
		$- 10$	$35$
	$2$	$- 7$

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.

$- 5$	$2$	$3$	$- 13$	$- 15$
		$- 10$	$35$
	$2$	$- 7$	$22$

Step 8.7

Multiply the newest entry in the result $(22)$ by the divisor $(- 5)$ and place the result of $(- 110)$ under the next term in the dividend $(- 15)$ .

$- 5$	$2$	$3$	$- 13$	$- 15$
		$- 10$	$35$	$- 110$
	$2$	$- 7$	$22$

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.

$- 5$	$2$	$3$	$- 13$	$- 15$
		$- 10$	$35$	$- 110$
	$2$	$- 7$	$22$	$- 125$

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.

$2 x^{2} + - 7 x + 22 + \frac{- 125}{x + 5}$

Step 8.10

Simplify the quotient polynomial.

$2 x^{2} - 7 x + 22 - \frac{125}{x + 5}$

Step 9

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 10

Apply synthetic division on $\frac{2 x^{3} + 3 x^{2} - 13 x - 15}{x - \frac{5}{2}}$ 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.

$\frac{5}{2}$	$2$	$3$	$- 13$	$- 15$

Step 10.2

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

$\frac{5}{2}$	$2$	$3$	$- 13$	$- 15$

	$2$

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 $(3)$ .

$\frac{5}{2}$	$2$	$3$	$- 13$	$- 15$
		$5$
	$2$

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.

$\frac{5}{2}$	$2$	$3$	$- 13$	$- 15$
		$5$
	$2$	$8$

Step 10.5

Multiply the newest entry in the result $(8)$ by the divisor $(\frac{5}{2})$ and place the result of $(20)$ under the next term in the dividend $(- 13)$ .

$\frac{5}{2}$	$2$	$3$	$- 13$	$- 15$
		$5$	$20$
	$2$	$8$

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.

$\frac{5}{2}$	$2$	$3$	$- 13$	$- 15$
		$5$	$20$
	$2$	$8$	$7$

Step 10.7

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 $(- 15)$ .

$\frac{5}{2}$	$2$	$3$	$- 13$	$- 15$
		$5$	$20$	$\frac{35}{2}$
	$2$	$8$	$7$

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.

$\frac{5}{2}$	$2$	$3$	$- 13$	$- 15$
		$5$	$20$	$\frac{35}{2}$
	$2$	$8$	$7$	$\frac{5}{2}$

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.

$2 x^{2} + 8 x + 7 + \frac{\frac{5}{2}}{x - \frac{5}{2}}$

Step 10.10

Simplify the quotient polynomial.

$2 x^{2} + 8 x + 7 + \frac{5}{2 x - 5}$

Step 11

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 12

Apply synthetic division on $\frac{2 x^{3} + 3 x^{2} - 13 x - 15}{x - 15}$ when $x = 15$ .

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Step 12.1

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

$15$	$2$	$3$	$- 13$	$- 15$

Step 12.2

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

$15$	$2$	$3$	$- 13$	$- 15$

	$2$

Step 12.3

Multiply the newest entry in the result $(2)$ by the divisor $(15)$ and place the result of $(30)$ under the next term in the dividend $(3)$ .

$15$	$2$	$3$	$- 13$	$- 15$
		$30$
	$2$

Step 12.4

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

$15$	$2$	$3$	$- 13$	$- 15$
		$30$
	$2$	$33$

Step 12.5

Multiply the newest entry in the result $(33)$ by the divisor $(15)$ and place the result of $(495)$ under the next term in the dividend $(- 13)$ .

$15$	$2$	$3$	$- 13$	$- 15$
		$30$	$495$
	$2$	$33$

Step 12.6

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

$15$	$2$	$3$	$- 13$	$- 15$
		$30$	$495$
	$2$	$33$	$482$

Step 12.7

Multiply the newest entry in the result $(482)$ by the divisor $(15)$ and place the result of $(7230)$ under the next term in the dividend $(- 15)$ .

$15$	$2$	$3$	$- 13$	$- 15$
		$30$	$495$	$7230$
	$2$	$33$	$482$

Step 12.8

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

$15$	$2$	$3$	$- 13$	$- 15$
		$30$	$495$	$7230$
	$2$	$33$	$482$	$7215$

Step 12.9

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

$2 x^{2} + 33 x + 482 + \frac{7215}{x - 15}$

Step 13

Since $15 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $15$ is an upper bound for the real roots of the function.

Upper Bound: $15$

Step 14

Apply synthetic division on $\frac{2 x^{3} + 3 x^{2} - 13 x - 15}{x + 15}$ when $x = - 15$ .

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Step 14.1

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

$- 15$	$2$	$3$	$- 13$	$- 15$

Step 14.2

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

$- 15$	$2$	$3$	$- 13$	$- 15$

	$2$

Step 14.3

Multiply the newest entry in the result $(2)$ by the divisor $(- 15)$ and place the result of $(- 30)$ under the next term in the dividend $(3)$ .

$- 15$	$2$	$3$	$- 13$	$- 15$
		$- 30$
	$2$

Step 14.4

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

$- 15$	$2$	$3$	$- 13$	$- 15$
		$- 30$
	$2$	$- 27$

Step 14.5

Multiply the newest entry in the result $(- 27)$ by the divisor $(- 15)$ and place the result of $(405)$ under the next term in the dividend $(- 13)$ .

$- 15$	$2$	$3$	$- 13$	$- 15$
		$- 30$	$405$
	$2$	$- 27$

Step 14.6

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

$- 15$	$2$	$3$	$- 13$	$- 15$
		$- 30$	$405$
	$2$	$- 27$	$392$

Step 14.7

Multiply the newest entry in the result $(392)$ by the divisor $(- 15)$ and place the result of $(- 5880)$ under the next term in the dividend $(- 15)$ .

$- 15$	$2$	$3$	$- 13$	$- 15$
		$- 30$	$405$	$- 5880$
	$2$	$- 27$	$392$

Step 14.8

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

$- 15$	$2$	$3$	$- 13$	$- 15$
		$- 30$	$405$	$- 5880$
	$2$	$- 27$	$392$	$- 5895$

Step 14.9

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

$2 x^{2} + - 27 x + 392 + \frac{- 5895}{x + 15}$

Step 14.10

Simplify the quotient polynomial.

$2 x^{2} - 27 x + 392 - \frac{5895}{x + 15}$

Step 15

Since $- 15 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 15$ is a lower bound for the real roots of the function.

Lower Bound: $- 15$

Step 16

Apply synthetic division on $\frac{2 x^{3} + 3 x^{2} - 13 x - 15}{x - \frac{15}{2}}$ when $x = \frac{15}{2}$ .

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Step 16.1

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

$\frac{15}{2}$	$2$	$3$	$- 13$	$- 15$

Step 16.2

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

$\frac{15}{2}$	$2$	$3$	$- 13$	$- 15$

	$2$

Step 16.3

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

$\frac{15}{2}$	$2$	$3$	$- 13$	$- 15$
		$15$
	$2$

Step 16.4

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

$\frac{15}{2}$	$2$	$3$	$- 13$	$- 15$
		$15$
	$2$	$18$

Step 16.5

Multiply the newest entry in the result $(18)$ by the divisor $(\frac{15}{2})$ and place the result of $(135)$ under the next term in the dividend $(- 13)$ .

$\frac{15}{2}$	$2$	$3$	$- 13$	$- 15$
		$15$	$135$
	$2$	$18$

Step 16.6

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

$\frac{15}{2}$	$2$	$3$	$- 13$	$- 15$
		$15$	$135$
	$2$	$18$	$122$

Step 16.7

Multiply the newest entry in the result $(122)$ by the divisor $(\frac{15}{2})$ and place the result of $(915)$ under the next term in the dividend $(- 15)$ .

$\frac{15}{2}$	$2$	$3$	$- 13$	$- 15$
		$15$	$135$	$915$
	$2$	$18$	$122$

Step 16.8

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

$\frac{15}{2}$	$2$	$3$	$- 13$	$- 15$
		$15$	$135$	$915$
	$2$	$18$	$122$	$900$

Step 16.9

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

$2 x^{2} + 18 x + 122 + \frac{900}{x - \frac{15}{2}}$

Step 16.10

Simplify the quotient polynomial.

$2 x^{2} + 18 x + 122 + \frac{1800}{2 x - 15}$

Step 17

Since $\frac{15}{2} > 0$ and all of the signs in the bottom row of the synthetic division are positive, $\frac{15}{2}$ is an upper bound for the real roots of the function.

Upper Bound: $\frac{15}{2}$

Step 18

Apply synthetic division on $\frac{2 x^{3} + 3 x^{2} - 13 x - 15}{x + \frac{15}{2}}$ when $x = - \frac{15}{2}$ .

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Step 18.1

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

$- \frac{15}{2}$	$2$	$3$	$- 13$	$- 15$

Step 18.2

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

$- \frac{15}{2}$	$2$	$3$	$- 13$	$- 15$

	$2$

Step 18.3

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

$- \frac{15}{2}$	$2$	$3$	$- 13$	$- 15$
		$- 15$
	$2$

Step 18.4

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

$- \frac{15}{2}$	$2$	$3$	$- 13$	$- 15$
		$- 15$
	$2$	$- 12$

Step 18.5

Multiply the newest entry in the result $(- 12)$ by the divisor $(- \frac{15}{2})$ and place the result of $(90)$ under the next term in the dividend $(- 13)$ .

$- \frac{15}{2}$	$2$	$3$	$- 13$	$- 15$
		$- 15$	$90$
	$2$	$- 12$

Step 18.6

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

$- \frac{15}{2}$	$2$	$3$	$- 13$	$- 15$
		$- 15$	$90$
	$2$	$- 12$	$77$

Step 18.7

Multiply the newest entry in the result $(77)$ by the divisor $(- \frac{15}{2})$ and place the result of $(- \frac{1155}{2})$ under the next term in the dividend $(- 15)$ .

$- \frac{15}{2}$	$2$	$3$	$- 13$	$- 15$
		$- 15$	$90$	$- \frac{1155}{2}$
	$2$	$- 12$	$77$

Step 18.8

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

$- \frac{15}{2}$	$2$	$3$	$- 13$	$- 15$
		$- 15$	$90$	$- \frac{1155}{2}$
	$2$	$- 12$	$77$	$- \frac{1185}{2}$

Step 18.9

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

$2 x^{2} + - 12 x + 77 + \frac{- \frac{1185}{2}}{x + \frac{15}{2}}$

Step 18.10

Simplify the quotient polynomial.

$2 x^{2} - 12 x + 77 - \frac{1185}{2 x + 15}$

Step 19

Since $- \frac{15}{2} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{15}{2}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{15}{2}$

Step 20

Determine the upper and lower bounds.

Upper Bounds: $3, 5, \frac{5}{2}, 15, \frac{15}{2}$

Lower Bounds: $- \frac{3}{2}, - 5, - 15, - \frac{15}{2}$

Step 21