Precalculus Examples

p (x) = 4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9

$p (x) = 4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9$

Step 1

Find every combination of

\pm \frac{p}{q}

$\pm \frac{p}{q}$ .

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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.

$p = \pm 1, \pm 3, \pm 9$

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

Step 1.2

Find every combination of

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm 9, \pm \frac{9}{2}, \pm \frac{9}{4}$

Step 2

Apply synthetic division on

$\frac{4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9}{x - 3}$ when

$x = 3$ .

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Step 2.1

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

$3$	$4$	$8$	$- 7$	$- 21$	$- 9$

Step 2.2

The first number in the dividend

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

$3$	$4$	$8$	$- 7$	$- 21$	$- 9$

	$4$

Step 2.3

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

$(8)$ .

$3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$12$
	$4$

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.

$3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$12$
	$4$	$20$

Step 2.5

Multiply the newest entry in the result

$(20)$ by the divisor

$(3)$ and place the result of

$(60)$ under the next term in the dividend

$(- 7)$ .

$3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$12$	$60$
	$4$	$20$

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.

$3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$12$	$60$
	$4$	$20$	$53$

Step 2.7

Multiply the newest entry in the result

$(53)$ by the divisor

$(3)$ and place the result of

$(159)$ under the next term in the dividend

$(- 21)$ .

$3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$12$	$60$	$159$
	$4$	$20$	$53$

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.

$3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$12$	$60$	$159$
	$4$	$20$	$53$	$138$

Step 2.9

Multiply the newest entry in the result

$(138)$ by the divisor

$(3)$ and place the result of

$(414)$ under the next term in the dividend

$(- 9)$ .

$3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$12$	$60$	$159$	$414$
	$4$	$20$	$53$	$138$

Step 2.10

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

$3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$12$	$60$	$159$	$414$
	$4$	$20$	$53$	$138$	$405$

Step 2.11

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

$4 x^{3} + 20 x^{2} + (53) x + 138 + \frac{405}{x - 3}$

Step 2.12

Simplify the quotient polynomial.

$4 x^{3} + 20 x^{2} + 53 x + 138 + \frac{405}{x - 3}$

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

$\frac{4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9}{x + 3}$ when

$x = - 3$ .

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Step 4.1

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

$- 3$	$4$	$8$	$- 7$	$- 21$	$- 9$

Step 4.2

The first number in the dividend

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

$- 3$	$4$	$8$	$- 7$	$- 21$	$- 9$

	$4$

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

$(8)$ .

$- 3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 12$
	$4$

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.

$- 3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 12$
	$4$	$- 4$

Step 4.5

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

$(- 7)$ .

$- 3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 12$	$12$
	$4$	$- 4$

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.

$- 3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 12$	$12$
	$4$	$- 4$	$5$

Step 4.7

Multiply the newest entry in the result

$(5)$ by the divisor

$(- 3)$ and place the result of

$(- 15)$ under the next term in the dividend

$(- 21)$ .

$- 3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 12$	$12$	$- 15$
	$4$	$- 4$	$5$

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.

$- 3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 12$	$12$	$- 15$
	$4$	$- 4$	$5$	$- 36$

Step 4.9

Multiply the newest entry in the result

$(- 36)$ by the divisor

$(- 3)$ and place the result of

$(108)$ under the next term in the dividend

$(- 9)$ .

$- 3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 12$	$12$	$- 15$	$108$
	$4$	$- 4$	$5$	$- 36$

Step 4.10

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

$- 3$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 12$	$12$	$- 15$	$108$
	$4$	$- 4$	$5$	$- 36$	$99$

Step 4.11

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

$4 x^{3} + - 4 x^{2} + (5) x - 36 + \frac{99}{x + 3}$

Step 4.12

Simplify the quotient polynomial.

$4 x^{3} - 4 x^{2} + 5 x - 36 + \frac{99}{x + 3}$

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

$\frac{4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9}{x - 9}$ when

$x = 9$ .

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Step 6.1

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

$9$	$4$	$8$	$- 7$	$- 21$	$- 9$

Step 6.2

The first number in the dividend

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

$9$	$4$	$8$	$- 7$	$- 21$	$- 9$

	$4$

Step 6.3

Multiply the newest entry in the result

$(4)$ by the divisor

$(9)$ and place the result of

$(36)$ under the next term in the dividend

$(8)$ .

$9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$36$
	$4$

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.

$9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$36$
	$4$	$44$

Step 6.5

Multiply the newest entry in the result

$(44)$ by the divisor

$(9)$ and place the result of

$(396)$ under the next term in the dividend

$(- 7)$ .

$9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$36$	$396$
	$4$	$44$

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.

$9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$36$	$396$
	$4$	$44$	$389$

Step 6.7

Multiply the newest entry in the result

$(389)$ by the divisor

$(9)$ and place the result of

$(3501)$ under the next term in the dividend

$(- 21)$ .

$9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$36$	$396$	$3501$
	$4$	$44$	$389$

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.

$9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$36$	$396$	$3501$
	$4$	$44$	$389$	$3480$

Step 6.9

Multiply the newest entry in the result

$(3480)$ by the divisor

$(9)$ and place the result of

$(31320)$ under the next term in the dividend

$(- 9)$ .

$9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$36$	$396$	$3501$	$31320$
	$4$	$44$	$389$	$3480$

Step 6.10

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

$9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$36$	$396$	$3501$	$31320$
	$4$	$44$	$389$	$3480$	$31311$

Step 6.11

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

$4 x^{3} + 44 x^{2} + (389) x + 3480 + \frac{31311}{x - 9}$

Step 6.12

Simplify the quotient polynomial.

$4 x^{3} + 44 x^{2} + 389 x + 3480 + \frac{31311}{x - 9}$

Step 7

Since

$9 > 0$ and all of the signs in the bottom row of the synthetic division are positive,

$9$ is an upper bound for the real roots of the function.

Upper Bound:

$9$

Step 8

Apply synthetic division on

$\frac{4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9}{x + 9}$ when

$x = - 9$ .

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Step 8.1

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

$- 9$	$4$	$8$	$- 7$	$- 21$	$- 9$

Step 8.2

The first number in the dividend

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

$- 9$	$4$	$8$	$- 7$	$- 21$	$- 9$

	$4$

Step 8.3

Multiply the newest entry in the result

$(4)$ by the divisor

$(- 9)$ and place the result of

$(- 36)$ under the next term in the dividend

$(8)$ .

$- 9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 36$
	$4$

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.

$- 9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 36$
	$4$	$- 28$

Step 8.5

Multiply the newest entry in the result

$(- 28)$ by the divisor

$(- 9)$ and place the result of

$(252)$ under the next term in the dividend

$(- 7)$ .

$- 9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 36$	$252$
	$4$	$- 28$

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.

$- 9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 36$	$252$
	$4$	$- 28$	$245$

Step 8.7

Multiply the newest entry in the result

$(245)$ by the divisor

$(- 9)$ and place the result of

$(- 2205)$ under the next term in the dividend

$(- 21)$ .

$- 9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 36$	$252$	$- 2205$
	$4$	$- 28$	$245$

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.

$- 9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 36$	$252$	$- 2205$
	$4$	$- 28$	$245$	$- 2226$

Step 8.9

Multiply the newest entry in the result

$(- 2226)$ by the divisor

$(- 9)$ and place the result of

$(20034)$ under the next term in the dividend

$(- 9)$ .

$- 9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 36$	$252$	$- 2205$	$20034$
	$4$	$- 28$	$245$	$- 2226$

Step 8.10

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

$- 9$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 36$	$252$	$- 2205$	$20034$
	$4$	$- 28$	$245$	$- 2226$	$20025$

Step 8.11

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

$4 x^{3} + - 28 x^{2} + (245) x - 2226 + \frac{20025}{x + 9}$

Step 8.12

Simplify the quotient polynomial.

$4 x^{3} - 28 x^{2} + 245 x - 2226 + \frac{20025}{x + 9}$

Step 9

Since

$- 9 < 0$ and the signs in the bottom row of the synthetic division alternate sign,

$- 9$ is a lower bound for the real roots of the function.

Lower Bound:

$- 9$

Step 10

Apply synthetic division on

$\frac{4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9}{x - \frac{9}{2}}$ when

$x = \frac{9}{2}$ .

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Step 10.1

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

$\frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$

Step 10.2

The first number in the dividend

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

$\frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$

	$4$

Step 10.3

Multiply the newest entry in the result

$(4)$ by the divisor

$(\frac{9}{2})$ and place the result of

$(18)$ under the next term in the dividend

$(8)$ .

$\frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$18$
	$4$

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.

$\frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$18$
	$4$	$26$

Step 10.5

Multiply the newest entry in the result

$(26)$ by the divisor

$(\frac{9}{2})$ and place the result of

$(117)$ under the next term in the dividend

$(- 7)$ .

$\frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$18$	$117$
	$4$	$26$

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.

$\frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$18$	$117$
	$4$	$26$	$110$

Step 10.7

Multiply the newest entry in the result

$(110)$ by the divisor

$(\frac{9}{2})$ and place the result of

$(495)$ under the next term in the dividend

$(- 21)$ .

$\frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$18$	$117$	$495$
	$4$	$26$	$110$

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.

$\frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$18$	$117$	$495$
	$4$	$26$	$110$	$474$

Step 10.9

Multiply the newest entry in the result

$(474)$ by the divisor

$(\frac{9}{2})$ and place the result of

$(2133)$ under the next term in the dividend

$(- 9)$ .

$\frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$18$	$117$	$495$	$2133$
	$4$	$26$	$110$	$474$

Step 10.10

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

$\frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$18$	$117$	$495$	$2133$
	$4$	$26$	$110$	$474$	$2124$

Step 10.11

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

$4 x^{3} + 26 x^{2} + (110) x + 474 + \frac{2124}{x - \frac{9}{2}}$

Step 10.12

Simplify the quotient polynomial.

$4 x^{3} + 26 x^{2} + 110 x + 474 + \frac{4248}{2 x - 9}$

Step 11

Since

$\frac{9}{2} > 0$ and all of the signs in the bottom row of the synthetic division are positive,

$\frac{9}{2}$ is an upper bound for the real roots of the function.

Upper Bound:

$\frac{9}{2}$

Step 12

Apply synthetic division on

$\frac{4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9}{x + \frac{9}{2}}$ when

$x = - \frac{9}{2}$ .

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Step 12.1

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

$- \frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$

Step 12.2

The first number in the dividend

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

$- \frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$

	$4$

Step 12.3

Multiply the newest entry in the result

$(4)$ by the divisor

$(- \frac{9}{2})$ and place the result of

$(- 18)$ under the next term in the dividend

$(8)$ .

$- \frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 18$
	$4$

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.

$- \frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 18$
	$4$	$- 10$

Step 12.5

Multiply the newest entry in the result

$(- 10)$ by the divisor

$(- \frac{9}{2})$ and place the result of

$(45)$ under the next term in the dividend

$(- 7)$ .

$- \frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 18$	$45$
	$4$	$- 10$

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.

$- \frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 18$	$45$
	$4$	$- 10$	$38$

Step 12.7

Multiply the newest entry in the result

$(38)$ by the divisor

$(- \frac{9}{2})$ and place the result of

$(- 171)$ under the next term in the dividend

$(- 21)$ .

$- \frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 18$	$45$	$- 171$
	$4$	$- 10$	$38$

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.

$- \frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 18$	$45$	$- 171$
	$4$	$- 10$	$38$	$- 192$

Step 12.9

Multiply the newest entry in the result

$(- 192)$ by the divisor

$(- \frac{9}{2})$ and place the result of

$(864)$ under the next term in the dividend

$(- 9)$ .

$- \frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 18$	$45$	$- 171$	$864$
	$4$	$- 10$	$38$	$- 192$

Step 12.10

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

$- \frac{9}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 18$	$45$	$- 171$	$864$
	$4$	$- 10$	$38$	$- 192$	$855$

Step 12.11

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

$4 x^{3} + - 10 x^{2} + (38) x - 192 + \frac{855}{x + \frac{9}{2}}$

Step 12.12

Simplify the quotient polynomial.

$4 x^{3} - 10 x^{2} + 38 x - 192 + \frac{1710}{2 x + 9}$

Step 13

Since

$- \frac{9}{2} < 0$ and the signs in the bottom row of the synthetic division alternate sign,

$- \frac{9}{2}$ is a lower bound for the real roots of the function.

Lower Bound:

$- \frac{9}{2}$

Step 14

Apply synthetic division on

$\frac{4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9}{x - \frac{9}{4}}$ when

$x = \frac{9}{4}$ .

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Step 14.1

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

$\frac{9}{4}$	$4$	$8$	$- 7$	$- 21$	$- 9$

Step 14.2

The first number in the dividend

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

$\frac{9}{4}$	$4$	$8$	$- 7$	$- 21$	$- 9$

	$4$

Step 14.3

Multiply the newest entry in the result

$(4)$ by the divisor

$(\frac{9}{4})$ and place the result of

$(9)$ under the next term in the dividend

$(8)$ .

$\frac{9}{4}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$9$
	$4$

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.

$\frac{9}{4}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$9$
	$4$	$17$

Step 14.5

Multiply the newest entry in the result

$(17)$ by the divisor

$(\frac{9}{4})$ and place the result of

$(\frac{153}{4})$ under the next term in the dividend

$(- 7)$ .

$\frac{9}{4}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$9$	$\frac{153}{4}$
	$4$	$17$

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.

$\frac{9}{4}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$9$	$\frac{153}{4}$
	$4$	$17$	$\frac{125}{4}$

Step 14.7

Multiply the newest entry in the result

$(\frac{125}{4})$ by the divisor

$(\frac{9}{4})$ and place the result of

$(\frac{1125}{16})$ under the next term in the dividend

$(- 21)$ .

$\frac{9}{4}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$9$	$\frac{153}{4}$	$\frac{1125}{16}$
	$4$	$17$	$\frac{125}{4}$

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.

$\frac{9}{4}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$9$	$\frac{153}{4}$	$\frac{1125}{16}$
	$4$	$17$	$\frac{125}{4}$	$\frac{789}{16}$

Step 14.9

Multiply the newest entry in the result

$(\frac{789}{16})$ by the divisor

$(\frac{9}{4})$ and place the result of

$(\frac{7101}{64})$ under the next term in the dividend

$(- 9)$ .

$\frac{9}{4}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$9$	$\frac{153}{4}$	$\frac{1125}{16}$	$\frac{7101}{64}$
	$4$	$17$	$\frac{125}{4}$	$\frac{789}{16}$

Step 14.10

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

$\frac{9}{4}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$9$	$\frac{153}{4}$	$\frac{1125}{16}$	$\frac{7101}{64}$
	$4$	$17$	$\frac{125}{4}$	$\frac{789}{16}$	$\frac{6525}{64}$

Step 14.11

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

$4 x^{3} + 17 x^{2} + (\frac{125}{4}) x + \frac{789}{16} + \frac{\frac{6525}{64}}{x - \frac{9}{4}}$

Step 14.12

Simplify the quotient polynomial.

$4 x^{3} + 17 x^{2} + \frac{125 x}{4} + \frac{789}{16} + \frac{6525}{16 (4 x - 9)}$

Step 15

Since

$\frac{9}{4} > 0$ and all of the signs in the bottom row of the synthetic division are positive,

$\frac{9}{4}$ is an upper bound for the real roots of the function.

Upper Bound:

$\frac{9}{4}$

Step 16

Determine the upper and lower bounds.

Upper Bounds:

$3, 9, \frac{9}{2}, \frac{9}{4}$

Lower Bounds:

$- 3, - 9, - \frac{9}{2}$

Step 17