Find the Upper and Lower Bounds f(x)=-3x^3+20x^2-36x+16

$f (x) = - 3 x^{3} + 20 x^{2} - 36 x + 16$

Step 1

Reorder the polynomial $- 3 x^{3} + 20 x^{2} - 36 x + 16$ .

Apply the distributive property.

$- 3 x^{3} \cdot - 1 + 20 x^{2} \cdot - 1 - 36 x \cdot - 1 + 16 \cdot - 1$

Simplify.

Multiply $- 1$ by $- 3$ .

$3 x^{3} + 20 x^{2} \cdot - 1 - 36 x \cdot - 1 + 16 \cdot - 1$

Multiply $- 1$ by $20$ .

$3 x^{3} - 20 x^{2} - 36 x \cdot - 1 + 16 \cdot - 1$

Multiply $- 1$ by $- 36$ .

$3 x^{3} - 20 x^{2} + 36 x + 16 \cdot - 1$

Multiply $16$ by $- 1$ .

$3 x^{3} - 20 x^{2} + 36 x - 16$

Step 2

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

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.

$p = \pm 1, \pm 2, \pm 4, \pm 8, \pm 16$

$q = \pm 1, \pm 3$

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

$\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3}, \pm 8, \pm \frac{8}{3}, \pm 16, \pm \frac{16}{3}$

Step 3

Apply synthetic division on $\frac{- 3 x^{3} + 20 x^{2} - 36 x + 16}{x + 1}$ when $x = - 1$ .

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

$- 1$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0

The first number in the dividend $(- 3)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0
	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 1)$ and place the result of $(3)$ under the next term in the dividend $(20)$ .

$- 1$	$- 3$	$20$	$- 36$	$16$
	\xa0	$3$		\xa0
	$- 3$

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

$- 1$	$- 3$	$20$	$- 36$	$16$
	\xa0	$3$		\xa0
	$- 3$	$23$

Multiply the newest entry in the result $(23)$ by the divisor $(- 1)$ and place the result of $(- 23)$ under the next term in the dividend $(- 36)$ .

$- 1$	$- 3$	$20$	$- 36$	$16$
	\xa0	$3$	$- 23$	\xa0
	$- 3$	$23$

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

$- 1$	$- 3$	$20$	$- 36$	$16$
	\xa0	$3$	$- 23$	\xa0
	$- 3$	$23$	$- 59$

Multiply the newest entry in the result $(- 59)$ by the divisor $(- 1)$ and place the result of $(59)$ under the next term in the dividend $(16)$ .

$- 1$	$- 3$	$20$	$- 36$	$16$
	\xa0	$3$	$- 23$	$59$
	$- 3$	$23$	$- 59$

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

$- 1$	$- 3$	$20$	$- 36$	$16$
	\xa0	$3$	$- 23$	$59$
	$- 3$	$23$	$- 59$	$75$

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

$- 3 x^{2} + 23 x - 59 + \frac{75}{x + 1}$

Step 4

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 5

Apply synthetic division on $\frac{- 3 x^{3} + 20 x^{2} - 36 x + 16}{x + \frac{1}{3}}$ when $x = - \frac{1}{3}$ .

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

$- \frac{1}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0

The first number in the dividend $(- 3)$ is put into the first position of the result area (below the horizontal line).

$- \frac{1}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0
	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- \frac{1}{3})$ and place the result of $(1)$ under the next term in the dividend $(20)$ .

$- \frac{1}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$1$		\xa0
	$- 3$

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

$- \frac{1}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$1$		\xa0
	$- 3$	$21$

Multiply the newest entry in the result $(21)$ by the divisor $(- \frac{1}{3})$ and place the result of $(- 7)$ under the next term in the dividend $(- 36)$ .

$- \frac{1}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$1$	$- 7$	\xa0
	$- 3$	$21$

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

$- \frac{1}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$1$	$- 7$	\xa0
	$- 3$	$21$	$- 43$

Multiply the newest entry in the result $(- 43)$ by the divisor $(- \frac{1}{3})$ and place the result of $(\frac{43}{3})$ under the next term in the dividend $(16)$ .

$- \frac{1}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$1$	$- 7$	$\frac{43}{3}$
	$- 3$	$21$	$- 43$

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

$- \frac{1}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$1$	$- 7$	$\frac{43}{3}$
	$- 3$	$21$	$- 43$	$\frac{91}{3}$

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

$- 3 x^{2} + 21 x - 43 + \frac{\frac{91}{3}}{x + \frac{1}{3}}$

Simplify the quotient polynomial.

$- 3 x^{2} + 21 x - 43 + \frac{91}{3 x + 1}$

Step 6

Since $- \frac{1}{3} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{1}{3}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{1}{3}$

Step 7

Apply synthetic division on $\frac{- 3 x^{3} + 20 x^{2} - 36 x + 16}{x + 2}$ when $x = - 2$ .

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

$- 2$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0

The first number in the dividend $(- 3)$ is put into the first position of the result area (below the horizontal line).

$- 2$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0
	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 2)$ and place the result of $(6)$ under the next term in the dividend $(20)$ .

$- 2$	$- 3$	$20$	$- 36$	$16$
	\xa0	$6$		\xa0
	$- 3$

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

$- 2$	$- 3$	$20$	$- 36$	$16$
	\xa0	$6$		\xa0
	$- 3$	$26$

Multiply the newest entry in the result $(26)$ by the divisor $(- 2)$ and place the result of $(- 52)$ under the next term in the dividend $(- 36)$ .

$- 2$	$- 3$	$20$	$- 36$	$16$
	\xa0	$6$	$- 52$	\xa0
	$- 3$	$26$

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

$- 2$	$- 3$	$20$	$- 36$	$16$
	\xa0	$6$	$- 52$	\xa0
	$- 3$	$26$	$- 88$

Multiply the newest entry in the result $(- 88)$ by the divisor $(- 2)$ and place the result of $(176)$ under the next term in the dividend $(16)$ .

$- 2$	$- 3$	$20$	$- 36$	$16$
	\xa0	$6$	$- 52$	$176$
	$- 3$	$26$	$- 88$

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

$- 2$	$- 3$	$20$	$- 36$	$16$
	\xa0	$6$	$- 52$	$176$
	$- 3$	$26$	$- 88$	$192$

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

$- 3 x^{2} + 26 x - 88 + \frac{192}{x + 2}$

Step 8

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

Lower Bound: $- 2$

Step 9

Apply synthetic division on $\frac{- 3 x^{3} + 20 x^{2} - 36 x + 16}{x + \frac{2}{3}}$ when $x = - \frac{2}{3}$ .

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

$- \frac{2}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0

The first number in the dividend $(- 3)$ is put into the first position of the result area (below the horizontal line).

$- \frac{2}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0
	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- \frac{2}{3})$ and place the result of $(2)$ under the next term in the dividend $(20)$ .

$- \frac{2}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$2$		\xa0
	$- 3$

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

$- \frac{2}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$2$		\xa0
	$- 3$	$22$

Multiply the newest entry in the result $(22)$ by the divisor $(- \frac{2}{3})$ and place the result of $(- \frac{44}{3})$ under the next term in the dividend $(- 36)$ .

$- \frac{2}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$2$	$- \frac{44}{3}$	\xa0
	$- 3$	$22$

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

$- \frac{2}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$2$	$- \frac{44}{3}$	\xa0
	$- 3$	$22$	$- \frac{152}{3}$

Multiply the newest entry in the result $(- \frac{152}{3})$ by the divisor $(- \frac{2}{3})$ and place the result of $(\frac{304}{9})$ under the next term in the dividend $(16)$ .

$- \frac{2}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$2$	$- \frac{44}{3}$	$\frac{304}{9}$
	$- 3$	$22$	$- \frac{152}{3}$

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

$- \frac{2}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$2$	$- \frac{44}{3}$	$\frac{304}{9}$
	$- 3$	$22$	$- \frac{152}{3}$	$\frac{448}{9}$

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

$- 3 x^{2} + 22 x - \frac{152}{3} + \frac{\frac{448}{9}}{x + \frac{2}{3}}$

Simplify the quotient polynomial.

$- 3 x^{2} + 22 x - \frac{152}{3} + \frac{448}{3 (3 x + 2)}$

Step 10

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

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

Step 11

Apply synthetic division on $\frac{- 3 x^{3} + 20 x^{2} - 36 x + 16}{x + 4}$ when $x = - 4$ .

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

$- 4$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0

The first number in the dividend $(- 3)$ is put into the first position of the result area (below the horizontal line).

$- 4$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0
	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 4)$ and place the result of $(12)$ under the next term in the dividend $(20)$ .

$- 4$	$- 3$	$20$	$- 36$	$16$
	\xa0	$12$		\xa0
	$- 3$

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

$- 4$	$- 3$	$20$	$- 36$	$16$
	\xa0	$12$		\xa0
	$- 3$	$32$

Multiply the newest entry in the result $(32)$ by the divisor $(- 4)$ and place the result of $(- 128)$ under the next term in the dividend $(- 36)$ .

$- 4$	$- 3$	$20$	$- 36$	$16$
	\xa0	$12$	$- 128$	\xa0
	$- 3$	$32$

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

$- 4$	$- 3$	$20$	$- 36$	$16$
	\xa0	$12$	$- 128$	\xa0
	$- 3$	$32$	$- 164$

Multiply the newest entry in the result $(- 164)$ by the divisor $(- 4)$ and place the result of $(656)$ under the next term in the dividend $(16)$ .

$- 4$	$- 3$	$20$	$- 36$	$16$
	\xa0	$12$	$- 128$	$656$
	$- 3$	$32$	$- 164$

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

$- 4$	$- 3$	$20$	$- 36$	$16$
	\xa0	$12$	$- 128$	$656$
	$- 3$	$32$	$- 164$	$672$

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

$- 3 x^{2} + 32 x - 164 + \frac{672}{x + 4}$

Step 12

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

Lower Bound: $- 4$

Step 13

Apply synthetic division on $\frac{- 3 x^{3} + 20 x^{2} - 36 x + 16}{x + \frac{4}{3}}$ when $x = - \frac{4}{3}$ .

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

$- \frac{4}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0

The first number in the dividend $(- 3)$ is put into the first position of the result area (below the horizontal line).

$- \frac{4}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0
	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- \frac{4}{3})$ and place the result of $(4)$ under the next term in the dividend $(20)$ .

$- \frac{4}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$4$		\xa0
	$- 3$

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

$- \frac{4}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$4$		\xa0
	$- 3$	$24$

Multiply the newest entry in the result $(24)$ by the divisor $(- \frac{4}{3})$ and place the result of $(- 32)$ under the next term in the dividend $(- 36)$ .

$- \frac{4}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$4$	$- 32$	\xa0
	$- 3$	$24$

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

$- \frac{4}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$4$	$- 32$	\xa0
	$- 3$	$24$	$- 68$

Multiply the newest entry in the result $(- 68)$ by the divisor $(- \frac{4}{3})$ and place the result of $(\frac{272}{3})$ under the next term in the dividend $(16)$ .

$- \frac{4}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$4$	$- 32$	$\frac{272}{3}$
	$- 3$	$24$	$- 68$

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

$- \frac{4}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$4$	$- 32$	$\frac{272}{3}$
	$- 3$	$24$	$- 68$	$\frac{320}{3}$

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

$- 3 x^{2} + 24 x - 68 + \frac{\frac{320}{3}}{x + \frac{4}{3}}$

Simplify the quotient polynomial.

$- 3 x^{2} + 24 x - 68 + \frac{320}{3 x + 4}$

Step 14

Since $- \frac{4}{3} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{4}{3}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{4}{3}$

Step 15

Apply synthetic division on $\frac{- 3 x^{3} + 20 x^{2} - 36 x + 16}{x + 8}$ when $x = - 8$ .

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

$- 8$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0

The first number in the dividend $(- 3)$ is put into the first position of the result area (below the horizontal line).

$- 8$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0
	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 8)$ and place the result of $(24)$ under the next term in the dividend $(20)$ .

$- 8$	$- 3$	$20$	$- 36$	$16$
	\xa0	$24$		\xa0
	$- 3$

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

$- 8$	$- 3$	$20$	$- 36$	$16$
	\xa0	$24$		\xa0
	$- 3$	$44$

Multiply the newest entry in the result $(44)$ by the divisor $(- 8)$ and place the result of $(- 352)$ under the next term in the dividend $(- 36)$ .

$- 8$	$- 3$	$20$	$- 36$	$16$
	\xa0	$24$	$- 352$	\xa0
	$- 3$	$44$

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

$- 8$	$- 3$	$20$	$- 36$	$16$
	\xa0	$24$	$- 352$	\xa0
	$- 3$	$44$	$- 388$

Multiply the newest entry in the result $(- 388)$ by the divisor $(- 8)$ and place the result of $(3104)$ under the next term in the dividend $(16)$ .

$- 8$	$- 3$	$20$	$- 36$	$16$
	\xa0	$24$	$- 352$	$3104$
	$- 3$	$44$	$- 388$

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

$- 8$	$- 3$	$20$	$- 36$	$16$
	\xa0	$24$	$- 352$	$3104$
	$- 3$	$44$	$- 388$	$3120$

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

$- 3 x^{2} + 44 x - 388 + \frac{3120}{x + 8}$

Step 16

Since $- 8 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 8$ is a lower bound for the real roots of the function.

Lower Bound: $- 8$

Step 17

Apply synthetic division on $\frac{- 3 x^{3} + 20 x^{2} - 36 x + 16}{x + \frac{8}{3}}$ when $x = - \frac{8}{3}$ .

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

$- \frac{8}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0

The first number in the dividend $(- 3)$ is put into the first position of the result area (below the horizontal line).

$- \frac{8}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0
	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- \frac{8}{3})$ and place the result of $(8)$ under the next term in the dividend $(20)$ .

$- \frac{8}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$8$		\xa0
	$- 3$

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

$- \frac{8}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$8$		\xa0
	$- 3$	$28$

Multiply the newest entry in the result $(28)$ by the divisor $(- \frac{8}{3})$ and place the result of $(- \frac{224}{3})$ under the next term in the dividend $(- 36)$ .

$- \frac{8}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$8$	$- \frac{224}{3}$	\xa0
	$- 3$	$28$

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

$- \frac{8}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$8$	$- \frac{224}{3}$	\xa0
	$- 3$	$28$	$- \frac{332}{3}$

Multiply the newest entry in the result $(- \frac{332}{3})$ by the divisor $(- \frac{8}{3})$ and place the result of $(\frac{2656}{9})$ under the next term in the dividend $(16)$ .

$- \frac{8}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$8$	$- \frac{224}{3}$	$\frac{2656}{9}$
	$- 3$	$28$	$- \frac{332}{3}$

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

$- \frac{8}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$8$	$- \frac{224}{3}$	$\frac{2656}{9}$
	$- 3$	$28$	$- \frac{332}{3}$	$\frac{2800}{9}$

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

$- 3 x^{2} + 28 x - \frac{332}{3} + \frac{\frac{2800}{9}}{x + \frac{8}{3}}$

Simplify the quotient polynomial.

$- 3 x^{2} + 28 x - \frac{332}{3} + \frac{2800}{3 (3 x + 8)}$

Step 18

Since $- \frac{8}{3} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{8}{3}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{8}{3}$

Step 19

Apply synthetic division on $\frac{- 3 x^{3} + 20 x^{2} - 36 x + 16}{x + 16}$ when $x = - 16$ .

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

$- 16$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0

The first number in the dividend $(- 3)$ is put into the first position of the result area (below the horizontal line).

$- 16$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0
	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 16)$ and place the result of $(48)$ under the next term in the dividend $(20)$ .

$- 16$	$- 3$	$20$	$- 36$	$16$
	\xa0	$48$		\xa0
	$- 3$

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

$- 16$	$- 3$	$20$	$- 36$	$16$
	\xa0	$48$		\xa0
	$- 3$	$68$

Multiply the newest entry in the result $(68)$ by the divisor $(- 16)$ and place the result of $(- 1088)$ under the next term in the dividend $(- 36)$ .

$- 16$	$- 3$	$20$	$- 36$	$16$
	\xa0	$48$	$- 1088$	\xa0
	$- 3$	$68$

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

$- 16$	$- 3$	$20$	$- 36$	$16$
	\xa0	$48$	$- 1088$	\xa0
	$- 3$	$68$	$- 1124$

Multiply the newest entry in the result $(- 1124)$ by the divisor $(- 16)$ and place the result of $(17984)$ under the next term in the dividend $(16)$ .

$- 16$	$- 3$	$20$	$- 36$	$16$
	\xa0	$48$	$- 1088$	$17984$
	$- 3$	$68$	$- 1124$

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

$- 16$	$- 3$	$20$	$- 36$	$16$
	\xa0	$48$	$- 1088$	$17984$
	$- 3$	$68$	$- 1124$	$18000$

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

$- 3 x^{2} + 68 x - 1124 + \frac{18000}{x + 16}$

Step 20

Since $- 16 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 16$ is a lower bound for the real roots of the function.

Lower Bound: $- 16$

Step 21

Apply synthetic division on $\frac{- 3 x^{3} + 20 x^{2} - 36 x + 16}{x + \frac{16}{3}}$ when $x = - \frac{16}{3}$ .

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

$- \frac{16}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0

The first number in the dividend $(- 3)$ is put into the first position of the result area (below the horizontal line).

$- \frac{16}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0			\xa0
	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- \frac{16}{3})$ and place the result of $(16)$ under the next term in the dividend $(20)$ .

$- \frac{16}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$16$		\xa0
	$- 3$

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

$- \frac{16}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$16$		\xa0
	$- 3$	$36$

Multiply the newest entry in the result $(36)$ by the divisor $(- \frac{16}{3})$ and place the result of $(- 192)$ under the next term in the dividend $(- 36)$ .

$- \frac{16}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$16$	$- 192$	\xa0
	$- 3$	$36$

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

$- \frac{16}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$16$	$- 192$	\xa0
	$- 3$	$36$	$- 228$

Multiply the newest entry in the result $(- 228)$ by the divisor $(- \frac{16}{3})$ and place the result of $(1216)$ under the next term in the dividend $(16)$ .

$- \frac{16}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$16$	$- 192$	$1216$
	$- 3$	$36$	$- 228$

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

$- \frac{16}{3}$	$- 3$	$20$	$- 36$	$16$
	\xa0	$16$	$- 192$	$1216$
	$- 3$	$36$	$- 228$	$1232$

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

$- 3 x^{2} + 36 x - 228 + \frac{1232}{x + \frac{16}{3}}$

Simplify the quotient polynomial.

$- 3 x^{2} + 36 x - 228 + \frac{3696}{3 x + 16}$

Step 22

Since $- \frac{16}{3} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{16}{3}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{16}{3}$

Step 23

Determine the upper and lower bounds.

No Upper Bounds

Lower Bounds: $- 1, - \frac{1}{3}, - 2, - \frac{2}{3}, - 4, - \frac{4}{3}, - 8, - \frac{8}{3}, - 16, - \frac{16}{3}$

Step 24

Find the Upper and Lower Bounds f(x)=-3x^3+20x^2-36x+16

Related Questions

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