Precalculus Examples

$f (x) = 4 x^{3} - 12 x^{2} - x + 15$

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.

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

$q = \pm 1, \pm 2, \pm 4$

Step 1.2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4}, \pm 15, \pm \frac{15}{2}, \pm \frac{15}{4}$

Step 2

Apply synthetic division on $\frac{4 x^{3} - 12 x^{2} - x + 15}{x + 1}$ when $x = - 1$ .

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Step 2.1

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

$- 1$	$4$	$- 12$	$- 1$	$15$

Step 2.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 2.3

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 $(- 12)$ .

$- 1$	$4$	$- 12$	$- 1$	$15$
		$- 4$
	$4$

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.

$- 1$	$4$	$- 12$	$- 1$	$15$
		$- 4$
	$4$	$- 16$

Step 2.5

Multiply the newest entry in the result $(- 16)$ by the divisor $(- 1)$ and place the result of $(16)$ under the next term in the dividend $(- 1)$ .

$- 1$	$4$	$- 12$	$- 1$	$15$
		$- 4$	$16$
	$4$	$- 16$

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.

$- 1$	$4$	$- 12$	$- 1$	$15$
		$- 4$	$16$
	$4$	$- 16$	$15$

Step 2.7

Multiply the newest entry in the result $(15)$ by the divisor $(- 1)$ and place the result of $(- 15)$ under the next term in the dividend $(15)$ .

$- 1$	$4$	$- 12$	$- 1$	$15$
		$- 4$	$16$	$- 15$
	$4$	$- 16$	$15$

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.

$- 1$	$4$	$- 12$	$- 1$	$15$
		$- 4$	$16$	$- 15$
	$4$	$- 16$	$15$	$0$

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.

$4 x^{2} + - 16 x + 15$

Step 2.10

Simplify the quotient polynomial.

$4 x^{2} - 16 x + 15$

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 $\frac{4 x^{3} - 12 x^{2} - x + 15}{x + 3}$ when $x = - 3$ .

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Step 4.1

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

$- 3$	$4$	$- 12$	$- 1$	$15$

Step 4.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- 3$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 4.3

Multiply the newest entry in the result $(4)$ by the divisor $(- 3)$ and place the result of $(- 12)$ under the next term in the dividend $(- 12)$ .

$- 3$	$4$	$- 12$	$- 1$	$15$
		$- 12$
	$4$

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.

$- 3$	$4$	$- 12$	$- 1$	$15$
		$- 12$
	$4$	$- 24$

Step 4.5

Multiply the newest entry in the result $(- 24)$ by the divisor $(- 3)$ and place the result of $(72)$ under the next term in the dividend $(- 1)$ .

$- 3$	$4$	$- 12$	$- 1$	$15$
		$- 12$	$72$
	$4$	$- 24$

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.

$- 3$	$4$	$- 12$	$- 1$	$15$
		$- 12$	$72$
	$4$	$- 24$	$71$

Step 4.7

Multiply the newest entry in the result $(71)$ by the divisor $(- 3)$ and place the result of $(- 213)$ under the next term in the dividend $(15)$ .

$- 3$	$4$	$- 12$	$- 1$	$15$
		$- 12$	$72$	$- 213$
	$4$	$- 24$	$71$

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.

$- 3$	$4$	$- 12$	$- 1$	$15$
		$- 12$	$72$	$- 213$
	$4$	$- 24$	$71$	$- 198$

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.

$4 x^{2} + - 24 x + 71 + \frac{- 198}{x + 3}$

Step 4.10

Simplify the quotient polynomial.

$4 x^{2} - 24 x + 71 - \frac{198}{x + 3}$

Step 5

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

Lower Bound: $- 3$

Step 6

Apply synthetic division on $\frac{4 x^{3} - 12 x^{2} - x + 15}{x + \frac{3}{2}}$ when $x = - \frac{3}{2}$ .

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Step 6.1

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

$- \frac{3}{2}$	$4$	$- 12$	$- 1$	$15$

Step 6.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- \frac{3}{2}$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 6.3

Multiply the newest entry in the result $(4)$ by the divisor $(- \frac{3}{2})$ and place the result of $(- 6)$ under the next term in the dividend $(- 12)$ .

$- \frac{3}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 6$
	$4$

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.

$- \frac{3}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 6$
	$4$	$- 18$

Step 6.5

Multiply the newest entry in the result $(- 18)$ by the divisor $(- \frac{3}{2})$ and place the result of $(27)$ under the next term in the dividend $(- 1)$ .

$- \frac{3}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 6$	$27$
	$4$	$- 18$

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.

$- \frac{3}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 6$	$27$
	$4$	$- 18$	$26$

Step 6.7

Multiply the newest entry in the result $(26)$ by the divisor $(- \frac{3}{2})$ and place the result of $(- 39)$ under the next term in the dividend $(15)$ .

$- \frac{3}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 6$	$27$	$- 39$
	$4$	$- 18$	$26$

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.

$- \frac{3}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 6$	$27$	$- 39$
	$4$	$- 18$	$26$	$- 24$

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.

$4 x^{2} + - 18 x + 26 + \frac{- 24}{x + \frac{3}{2}}$

Step 6.10

Simplify the quotient polynomial.

$4 x^{2} - 18 x + 26 - \frac{48}{2 x + 3}$

Step 7

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 8

Apply synthetic division on $\frac{4 x^{3} - 12 x^{2} - x + 15}{x - 5}$ when $x = 5$ .

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Step 8.1

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

$5$	$4$	$- 12$	$- 1$	$15$

Step 8.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$5$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 8.3

Multiply the newest entry in the result $(4)$ by the divisor $(5)$ and place the result of $(20)$ under the next term in the dividend $(- 12)$ .

$5$	$4$	$- 12$	$- 1$	$15$
		$20$
	$4$

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.

$5$	$4$	$- 12$	$- 1$	$15$
		$20$
	$4$	$8$

Step 8.5

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 $(- 1)$ .

$5$	$4$	$- 12$	$- 1$	$15$
		$20$	$40$
	$4$	$8$

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.

$5$	$4$	$- 12$	$- 1$	$15$
		$20$	$40$
	$4$	$8$	$39$

Step 8.7

Multiply the newest entry in the result $(39)$ by the divisor $(5)$ and place the result of $(195)$ under the next term in the dividend $(15)$ .

$5$	$4$	$- 12$	$- 1$	$15$
		$20$	$40$	$195$
	$4$	$8$	$39$

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.

$5$	$4$	$- 12$	$- 1$	$15$
		$20$	$40$	$195$
	$4$	$8$	$39$	$210$

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.

$4 x^{2} + 8 x + 39 + \frac{210}{x - 5}$

Step 9

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 10

Apply synthetic division on $\frac{4 x^{3} - 12 x^{2} - x + 15}{x + 5}$ when $x = - 5$ .

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Step 10.1

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

$- 5$	$4$	$- 12$	$- 1$	$15$

Step 10.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- 5$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 10.3

Multiply the newest entry in the result $(4)$ by the divisor $(- 5)$ and place the result of $(- 20)$ under the next term in the dividend $(- 12)$ .

$- 5$	$4$	$- 12$	$- 1$	$15$
		$- 20$
	$4$

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.

$- 5$	$4$	$- 12$	$- 1$	$15$
		$- 20$
	$4$	$- 32$

Step 10.5

Multiply the newest entry in the result $(- 32)$ by the divisor $(- 5)$ and place the result of $(160)$ under the next term in the dividend $(- 1)$ .

$- 5$	$4$	$- 12$	$- 1$	$15$
		$- 20$	$160$
	$4$	$- 32$

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.

$- 5$	$4$	$- 12$	$- 1$	$15$
		$- 20$	$160$
	$4$	$- 32$	$159$

Step 10.7

Multiply the newest entry in the result $(159)$ by the divisor $(- 5)$ and place the result of $(- 795)$ under the next term in the dividend $(15)$ .

$- 5$	$4$	$- 12$	$- 1$	$15$
		$- 20$	$160$	$- 795$
	$4$	$- 32$	$159$

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.

$- 5$	$4$	$- 12$	$- 1$	$15$
		$- 20$	$160$	$- 795$
	$4$	$- 32$	$159$	$- 780$

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.

$4 x^{2} + - 32 x + 159 + \frac{- 780}{x + 5}$

Step 10.10

Simplify the quotient polynomial.

$4 x^{2} - 32 x + 159 - \frac{780}{x + 5}$

Step 11

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 12

Apply synthetic division on $\frac{4 x^{3} - 12 x^{2} - x + 15}{x + \frac{5}{2}}$ when $x = - \frac{5}{2}$ .

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Step 12.1

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

$- \frac{5}{2}$	$4$	$- 12$	$- 1$	$15$

Step 12.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- \frac{5}{2}$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 12.3

Multiply the newest entry in the result $(4)$ by the divisor $(- \frac{5}{2})$ and place the result of $(- 10)$ under the next term in the dividend $(- 12)$ .

$- \frac{5}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 10$
	$4$

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.

$- \frac{5}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 10$
	$4$	$- 22$

Step 12.5

Multiply the newest entry in the result $(- 22)$ by the divisor $(- \frac{5}{2})$ and place the result of $(55)$ under the next term in the dividend $(- 1)$ .

$- \frac{5}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 10$	$55$
	$4$	$- 22$

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.

$- \frac{5}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 10$	$55$
	$4$	$- 22$	$54$

Step 12.7

Multiply the newest entry in the result $(54)$ by the divisor $(- \frac{5}{2})$ and place the result of $(- 135)$ under the next term in the dividend $(15)$ .

$- \frac{5}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 10$	$55$	$- 135$
	$4$	$- 22$	$54$

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.

$- \frac{5}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 10$	$55$	$- 135$
	$4$	$- 22$	$54$	$- 120$

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.

$4 x^{2} + - 22 x + 54 + \frac{- 120}{x + \frac{5}{2}}$

Step 12.10

Simplify the quotient polynomial.

$4 x^{2} - 22 x + 54 - \frac{240}{2 x + 5}$

Step 13

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 14

Apply synthetic division on $\frac{4 x^{3} - 12 x^{2} - x + 15}{x + \frac{5}{4}}$ when $x = - \frac{5}{4}$ .

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Step 14.1

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

$- \frac{5}{4}$	$4$	$- 12$	$- 1$	$15$

Step 14.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- \frac{5}{4}$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 14.3

Multiply the newest entry in the result $(4)$ by the divisor $(- \frac{5}{4})$ and place the result of $(- 5)$ under the next term in the dividend $(- 12)$ .

$- \frac{5}{4}$	$4$	$- 12$	$- 1$	$15$
		$- 5$
	$4$

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.

$- \frac{5}{4}$	$4$	$- 12$	$- 1$	$15$
		$- 5$
	$4$	$- 17$

Step 14.5

Multiply the newest entry in the result $(- 17)$ by the divisor $(- \frac{5}{4})$ and place the result of $(\frac{85}{4})$ under the next term in the dividend $(- 1)$ .

$- \frac{5}{4}$	$4$	$- 12$	$- 1$	$15$
		$- 5$	$\frac{85}{4}$
	$4$	$- 17$

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.

$- \frac{5}{4}$	$4$	$- 12$	$- 1$	$15$
		$- 5$	$\frac{85}{4}$
	$4$	$- 17$	$\frac{81}{4}$

Step 14.7

Multiply the newest entry in the result $(\frac{81}{4})$ by the divisor $(- \frac{5}{4})$ and place the result of $(- \frac{405}{16})$ under the next term in the dividend $(15)$ .

$- \frac{5}{4}$	$4$	$- 12$	$- 1$	$15$
		$- 5$	$\frac{85}{4}$	$- \frac{405}{16}$
	$4$	$- 17$	$\frac{81}{4}$

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.

$- \frac{5}{4}$	$4$	$- 12$	$- 1$	$15$
		$- 5$	$\frac{85}{4}$	$- \frac{405}{16}$
	$4$	$- 17$	$\frac{81}{4}$	$- \frac{165}{16}$

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.

$4 x^{2} + - 17 x + \frac{81}{4} + \frac{- \frac{165}{16}}{x + \frac{5}{4}}$

Step 14.10

Simplify the quotient polynomial.

$4 x^{2} - 17 x + \frac{81}{4} - \frac{165}{4 (4 x + 5)}$

Step 15

Since $- \frac{5}{4} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{5}{4}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{5}{4}$

Step 16

Apply synthetic division on $\frac{4 x^{3} - 12 x^{2} - x + 15}{x - 15}$ when $x = 15$ .

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Step 16.1

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

$15$	$4$	$- 12$	$- 1$	$15$

Step 16.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$15$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 16.3

Multiply the newest entry in the result $(4)$ by the divisor $(15)$ and place the result of $(60)$ under the next term in the dividend $(- 12)$ .

$15$	$4$	$- 12$	$- 1$	$15$
		$60$
	$4$

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.

$15$	$4$	$- 12$	$- 1$	$15$
		$60$
	$4$	$48$

Step 16.5

Multiply the newest entry in the result $(48)$ by the divisor $(15)$ and place the result of $(720)$ under the next term in the dividend $(- 1)$ .

$15$	$4$	$- 12$	$- 1$	$15$
		$60$	$720$
	$4$	$48$

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.

$15$	$4$	$- 12$	$- 1$	$15$
		$60$	$720$
	$4$	$48$	$719$

Step 16.7

Multiply the newest entry in the result $(719)$ by the divisor $(15)$ and place the result of $(10785)$ under the next term in the dividend $(15)$ .

$15$	$4$	$- 12$	$- 1$	$15$
		$60$	$720$	$10785$
	$4$	$48$	$719$

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.

$15$	$4$	$- 12$	$- 1$	$15$
		$60$	$720$	$10785$
	$4$	$48$	$719$	$10800$

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.

$4 x^{2} + 48 x + 719 + \frac{10800}{x - 15}$

Step 17

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 18

Apply synthetic division on $\frac{4 x^{3} - 12 x^{2} - x + 15}{x + 15}$ when $x = - 15$ .

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Step 18.1

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

$- 15$	$4$	$- 12$	$- 1$	$15$

Step 18.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- 15$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 18.3

Multiply the newest entry in the result $(4)$ by the divisor $(- 15)$ and place the result of $(- 60)$ under the next term in the dividend $(- 12)$ .

$- 15$	$4$	$- 12$	$- 1$	$15$
		$- 60$
	$4$

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.

$- 15$	$4$	$- 12$	$- 1$	$15$
		$- 60$
	$4$	$- 72$

Step 18.5

Multiply the newest entry in the result $(- 72)$ by the divisor $(- 15)$ and place the result of $(1080)$ under the next term in the dividend $(- 1)$ .

$- 15$	$4$	$- 12$	$- 1$	$15$
		$- 60$	$1080$
	$4$	$- 72$

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.

$- 15$	$4$	$- 12$	$- 1$	$15$
		$- 60$	$1080$
	$4$	$- 72$	$1079$

Step 18.7

Multiply the newest entry in the result $(1079)$ by the divisor $(- 15)$ and place the result of $(- 16185)$ under the next term in the dividend $(15)$ .

$- 15$	$4$	$- 12$	$- 1$	$15$
		$- 60$	$1080$	$- 16185$
	$4$	$- 72$	$1079$

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.

$- 15$	$4$	$- 12$	$- 1$	$15$
		$- 60$	$1080$	$- 16185$
	$4$	$- 72$	$1079$	$- 16170$

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.

$4 x^{2} + - 72 x + 1079 + \frac{- 16170}{x + 15}$

Step 18.10

Simplify the quotient polynomial.

$4 x^{2} - 72 x + 1079 - \frac{16170}{x + 15}$

Step 19

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 20

Apply synthetic division on $\frac{4 x^{3} - 12 x^{2} - x + 15}{x - \frac{15}{2}}$ when $x = \frac{15}{2}$ .

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Step 20.1

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

$\frac{15}{2}$	$4$	$- 12$	$- 1$	$15$

Step 20.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$\frac{15}{2}$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 20.3

Multiply the newest entry in the result $(4)$ by the divisor $(\frac{15}{2})$ and place the result of $(30)$ under the next term in the dividend $(- 12)$ .

$\frac{15}{2}$	$4$	$- 12$	$- 1$	$15$
		$30$
	$4$

Step 20.4

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

$\frac{15}{2}$	$4$	$- 12$	$- 1$	$15$
		$30$
	$4$	$18$

Step 20.5

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 $(- 1)$ .

$\frac{15}{2}$	$4$	$- 12$	$- 1$	$15$
		$30$	$135$
	$4$	$18$

Step 20.6

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

$\frac{15}{2}$	$4$	$- 12$	$- 1$	$15$
		$30$	$135$
	$4$	$18$	$134$

Step 20.7

Multiply the newest entry in the result $(134)$ by the divisor $(\frac{15}{2})$ and place the result of $(1005)$ under the next term in the dividend $(15)$ .

$\frac{15}{2}$	$4$	$- 12$	$- 1$	$15$
		$30$	$135$	$1005$
	$4$	$18$	$134$

Step 20.8

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

$\frac{15}{2}$	$4$	$- 12$	$- 1$	$15$
		$30$	$135$	$1005$
	$4$	$18$	$134$	$1020$

Step 20.9

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

$4 x^{2} + 18 x + 134 + \frac{1020}{x - \frac{15}{2}}$

Step 20.10

Simplify the quotient polynomial.

$4 x^{2} + 18 x + 134 + \frac{2040}{2 x - 15}$

Step 21

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 22

Apply synthetic division on $\frac{4 x^{3} - 12 x^{2} - x + 15}{x + \frac{15}{2}}$ when $x = - \frac{15}{2}$ .

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Step 22.1

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

$- \frac{15}{2}$	$4$	$- 12$	$- 1$	$15$

Step 22.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- \frac{15}{2}$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 22.3

Multiply the newest entry in the result $(4)$ by the divisor $(- \frac{15}{2})$ and place the result of $(- 30)$ under the next term in the dividend $(- 12)$ .

$- \frac{15}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 30$
	$4$

Step 22.4

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

$- \frac{15}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 30$
	$4$	$- 42$

Step 22.5

Multiply the newest entry in the result $(- 42)$ by the divisor $(- \frac{15}{2})$ and place the result of $(315)$ under the next term in the dividend $(- 1)$ .

$- \frac{15}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 30$	$315$
	$4$	$- 42$

Step 22.6

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

$- \frac{15}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 30$	$315$
	$4$	$- 42$	$314$

Step 22.7

Multiply the newest entry in the result $(314)$ by the divisor $(- \frac{15}{2})$ and place the result of $(- 2355)$ under the next term in the dividend $(15)$ .

$- \frac{15}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 30$	$315$	$- 2355$
	$4$	$- 42$	$314$

Step 22.8

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

$- \frac{15}{2}$	$4$	$- 12$	$- 1$	$15$
		$- 30$	$315$	$- 2355$
	$4$	$- 42$	$314$	$- 2340$

Step 22.9

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

$4 x^{2} + - 42 x + 314 + \frac{- 2340}{x + \frac{15}{2}}$

Step 22.10

Simplify the quotient polynomial.

$4 x^{2} - 42 x + 314 - \frac{4680}{2 x + 15}$

Step 23

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 24

Apply synthetic division on $\frac{4 x^{3} - 12 x^{2} - x + 15}{x - \frac{15}{4}}$ when $x = \frac{15}{4}$ .

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Step 24.1

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

$\frac{15}{4}$	$4$	$- 12$	$- 1$	$15$

Step 24.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$\frac{15}{4}$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 24.3

Multiply the newest entry in the result $(4)$ by the divisor $(\frac{15}{4})$ and place the result of $(15)$ under the next term in the dividend $(- 12)$ .

$\frac{15}{4}$	$4$	$- 12$	$- 1$	$15$
		$15$
	$4$

Step 24.4

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

$\frac{15}{4}$	$4$	$- 12$	$- 1$	$15$
		$15$
	$4$	$3$

Step 24.5

Multiply the newest entry in the result $(3)$ by the divisor $(\frac{15}{4})$ and place the result of $(\frac{45}{4})$ under the next term in the dividend $(- 1)$ .

$\frac{15}{4}$	$4$	$- 12$	$- 1$	$15$
		$15$	$\frac{45}{4}$
	$4$	$3$

Step 24.6

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

$\frac{15}{4}$	$4$	$- 12$	$- 1$	$15$
		$15$	$\frac{45}{4}$
	$4$	$3$	$\frac{41}{4}$

Step 24.7

Multiply the newest entry in the result $(\frac{41}{4})$ by the divisor $(\frac{15}{4})$ and place the result of $(\frac{615}{16})$ under the next term in the dividend $(15)$ .

$\frac{15}{4}$	$4$	$- 12$	$- 1$	$15$
		$15$	$\frac{45}{4}$	$\frac{615}{16}$
	$4$	$3$	$\frac{41}{4}$

Step 24.8

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

$\frac{15}{4}$	$4$	$- 12$	$- 1$	$15$
		$15$	$\frac{45}{4}$	$\frac{615}{16}$
	$4$	$3$	$\frac{41}{4}$	$\frac{855}{16}$

Step 24.9

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

$4 x^{2} + 3 x + \frac{41}{4} + \frac{\frac{855}{16}}{x - \frac{15}{4}}$

Step 24.10

Simplify the quotient polynomial.

$4 x^{2} + 3 x + \frac{41}{4} + \frac{855}{4 (4 x - 15)}$

Step 25

Since $\frac{15}{4} > 0$ and all of the signs in the bottom row of the synthetic division are positive, $\frac{15}{4}$ is an upper bound for the real roots of the function.

Upper Bound: $\frac{15}{4}$

Step 26

Apply synthetic division on $\frac{4 x^{3} - 12 x^{2} - x + 15}{x + \frac{15}{4}}$ when $x = - \frac{15}{4}$ .

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Step 26.1

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

$- \frac{15}{4}$	$4$	$- 12$	$- 1$	$15$

Step 26.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- \frac{15}{4}$	$4$	$- 12$	$- 1$	$15$

	$4$

Step 26.3

Multiply the newest entry in the result $(4)$ by the divisor $(- \frac{15}{4})$ and place the result of $(- 15)$ under the next term in the dividend $(- 12)$ .

$- \frac{15}{4}$	$4$	$- 12$	$- 1$	$15$
		$- 15$
	$4$

Step 26.4

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

$- \frac{15}{4}$	$4$	$- 12$	$- 1$	$15$
		$- 15$
	$4$	$- 27$

Step 26.5

Multiply the newest entry in the result $(- 27)$ by the divisor $(- \frac{15}{4})$ and place the result of $(\frac{405}{4})$ under the next term in the dividend $(- 1)$ .

$- \frac{15}{4}$	$4$	$- 12$	$- 1$	$15$
		$- 15$	$\frac{405}{4}$
	$4$	$- 27$

Step 26.6

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

$- \frac{15}{4}$	$4$	$- 12$	$- 1$	$15$
		$- 15$	$\frac{405}{4}$
	$4$	$- 27$	$\frac{401}{4}$

Step 26.7

Multiply the newest entry in the result $(\frac{401}{4})$ by the divisor $(- \frac{15}{4})$ and place the result of $(- \frac{6015}{16})$ under the next term in the dividend $(15)$ .

$- \frac{15}{4}$	$4$	$- 12$	$- 1$	$15$
		$- 15$	$\frac{405}{4}$	$- \frac{6015}{16}$
	$4$	$- 27$	$\frac{401}{4}$

Step 26.8

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

$- \frac{15}{4}$	$4$	$- 12$	$- 1$	$15$
		$- 15$	$\frac{405}{4}$	$- \frac{6015}{16}$
	$4$	$- 27$	$\frac{401}{4}$	$- \frac{5775}{16}$

Step 26.9

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

$4 x^{2} + - 27 x + \frac{401}{4} + \frac{- \frac{5775}{16}}{x + \frac{15}{4}}$

Step 26.10

Simplify the quotient polynomial.

$4 x^{2} - 27 x + \frac{401}{4} - \frac{5775}{4 (4 x + 15)}$

Step 27

Since $- \frac{15}{4} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{15}{4}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{15}{4}$

Step 28

Determine the upper and lower bounds.

Upper Bounds: $5, 15, \frac{15}{2}, \frac{15}{4}$

Lower Bounds: $- 1, - 3, - \frac{3}{2}, - 5, - \frac{5}{2}, - \frac{5}{4}, - 15, - \frac{15}{2}, - \frac{15}{4}$

Step 29