Find the Upper and Lower Bounds x^4-5x^3+20x-16

$x^{4} - 5 x^{3} + 20 x - 16$

Step 1

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.

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

$q = \pm 1$

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

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

Step 2

Apply synthetic division on $\frac{x^{4} - 5 x^{3} + 20 x - 16}{x + 2}$ when $x = - 2$ .

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

$- 2$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0				\xa0

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

$- 2$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 2)$ and place the result of $(- 2)$ under the next term in the dividend $(- 5)$ .

$- 2$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 2$			\xa0
	$1$

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

$- 2$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 2$			\xa0
	$1$	$- 7$

Multiply the newest entry in the result $(- 7)$ by the divisor $(- 2)$ and place the result of $(14)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 2$	$14$		\xa0
	$1$	$- 7$

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

$- 2$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 2$	$14$		\xa0
	$1$	$- 7$	$14$

Multiply the newest entry in the result $(14)$ by the divisor $(- 2)$ and place the result of $(- 28)$ under the next term in the dividend $(20)$ .

$- 2$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 2$	$14$	$- 28$	\xa0
	$1$	$- 7$	$14$

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

$- 2$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 2$	$14$	$- 28$	\xa0
	$1$	$- 7$	$14$	$- 8$

Multiply the newest entry in the result $(- 8)$ by the divisor $(- 2)$ and place the result of $(16)$ under the next term in the dividend $(- 16)$ .

$- 2$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 2$	$14$	$- 28$	$16$
	$1$	$- 7$	$14$	$- 8$

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

$- 2$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 2$	$14$	$- 28$	$16$
	$1$	$- 7$	$14$	$- 8$	$0$

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

$1 x^{3} + - 7 x^{2} + (14) x - 8$

Simplify the quotient polynomial.

$x^{3} - 7 x^{2} + 14 x - 8$

Step 3

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 4

Apply synthetic division on $\frac{x^{4} - 5 x^{3} + 20 x - 16}{x + 4}$ when $x = - 4$ .

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

$- 4$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0				\xa0

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

$- 4$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0				\xa0
	$1$

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

$- 4$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 4$			\xa0
	$1$

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

$- 4$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 4$			\xa0
	$1$	$- 9$

Multiply the newest entry in the result $(- 9)$ by the divisor $(- 4)$ and place the result of $(36)$ under the next term in the dividend $(0)$ .

$- 4$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 4$	$36$		\xa0
	$1$	$- 9$

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

$- 4$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 4$	$36$		\xa0
	$1$	$- 9$	$36$

Multiply the newest entry in the result $(36)$ by the divisor $(- 4)$ and place the result of $(- 144)$ under the next term in the dividend $(20)$ .

$- 4$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 4$	$36$	$- 144$	\xa0
	$1$	$- 9$	$36$

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

$- 4$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 4$	$36$	$- 144$	\xa0
	$1$	$- 9$	$36$	$- 124$

Multiply the newest entry in the result $(- 124)$ by the divisor $(- 4)$ and place the result of $(496)$ under the next term in the dividend $(- 16)$ .

$- 4$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 4$	$36$	$- 144$	$496$
	$1$	$- 9$	$36$	$- 124$

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

$- 4$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 4$	$36$	$- 144$	$496$
	$1$	$- 9$	$36$	$- 124$	$480$

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

$1 x^{3} + - 9 x^{2} + (36) x - 124 + \frac{480}{x + 4}$

Simplify the quotient polynomial.

$x^{3} - 9 x^{2} + 36 x - 124 + \frac{480}{x + 4}$

Step 5

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 6

Apply synthetic division on $\frac{x^{4} - 5 x^{3} + 20 x - 16}{x - 8}$ when $x = 8$ .

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

$8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0				\xa0

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

$8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0				\xa0
	$1$

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

$8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$8$			\xa0
	$1$

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

$8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$8$			\xa0
	$1$	$3$

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 $(0)$ .

$8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$8$	$24$		\xa0
	$1$	$3$

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

$8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$8$	$24$		\xa0
	$1$	$3$	$24$

Multiply the newest entry in the result $(24)$ by the divisor $(8)$ and place the result of $(192)$ under the next term in the dividend $(20)$ .

$8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$8$	$24$	$192$	\xa0
	$1$	$3$	$24$

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

$8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$8$	$24$	$192$	\xa0
	$1$	$3$	$24$	$212$

Multiply the newest entry in the result $(212)$ by the divisor $(8)$ and place the result of $(1696)$ under the next term in the dividend $(- 16)$ .

$8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$8$	$24$	$192$	$1696$
	$1$	$3$	$24$	$212$

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

$8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$8$	$24$	$192$	$1696$
	$1$	$3$	$24$	$212$	$1680$

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

$1 x^{3} + 3 x^{2} + (24) x + 212 + \frac{1680}{x - 8}$

Simplify the quotient polynomial.

$x^{3} + 3 x^{2} + 24 x + 212 + \frac{1680}{x - 8}$

Step 7

Since $8 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $8$ is an upper bound for the real roots of the function.

Upper Bound: $8$

Step 8

Apply synthetic division on $\frac{x^{4} - 5 x^{3} + 20 x - 16}{x + 8}$ when $x = - 8$ .

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

$- 8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0				\xa0

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

$- 8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0				\xa0
	$1$

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

$- 8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 8$			\xa0
	$1$

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

$- 8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 8$			\xa0
	$1$	$- 13$

Multiply the newest entry in the result $(- 13)$ by the divisor $(- 8)$ and place the result of $(104)$ under the next term in the dividend $(0)$ .

$- 8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 8$	$104$		\xa0
	$1$	$- 13$

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

$- 8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 8$	$104$		\xa0
	$1$	$- 13$	$104$

Multiply the newest entry in the result $(104)$ by the divisor $(- 8)$ and place the result of $(- 832)$ under the next term in the dividend $(20)$ .

$- 8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 8$	$104$	$- 832$	\xa0
	$1$	$- 13$	$104$

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

$- 8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 8$	$104$	$- 832$	\xa0
	$1$	$- 13$	$104$	$- 812$

Multiply the newest entry in the result $(- 812)$ by the divisor $(- 8)$ and place the result of $(6496)$ under the next term in the dividend $(- 16)$ .

$- 8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 8$	$104$	$- 832$	$6496$
	$1$	$- 13$	$104$	$- 812$

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

$- 8$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 8$	$104$	$- 832$	$6496$
	$1$	$- 13$	$104$	$- 812$	$6480$

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

$1 x^{3} + - 13 x^{2} + (104) x - 812 + \frac{6480}{x + 8}$

Simplify the quotient polynomial.

$x^{3} - 13 x^{2} + 104 x - 812 + \frac{6480}{x + 8}$

Step 9

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 10

Apply synthetic division on $\frac{x^{4} - 5 x^{3} + 20 x - 16}{x - 16}$ when $x = 16$ .

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

$16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0				\xa0

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

$16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(16)$ and place the result of $(16)$ under the next term in the dividend $(- 5)$ .

$16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$16$			\xa0
	$1$

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

$16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$16$			\xa0
	$1$	$11$

Multiply the newest entry in the result $(11)$ by the divisor $(16)$ and place the result of $(176)$ under the next term in the dividend $(0)$ .

$16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$16$	$176$		\xa0
	$1$	$11$

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

$16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$16$	$176$		\xa0
	$1$	$11$	$176$

Multiply the newest entry in the result $(176)$ by the divisor $(16)$ and place the result of $(2816)$ under the next term in the dividend $(20)$ .

$16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$16$	$176$	$2816$	\xa0
	$1$	$11$	$176$

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

$16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$16$	$176$	$2816$	\xa0
	$1$	$11$	$176$	$2836$

Multiply the newest entry in the result $(2836)$ by the divisor $(16)$ and place the result of $(45376)$ under the next term in the dividend $(- 16)$ .

$16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$16$	$176$	$2816$	$45376$
	$1$	$11$	$176$	$2836$

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

$16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$16$	$176$	$2816$	$45376$
	$1$	$11$	$176$	$2836$	$45360$

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

$1 x^{3} + 11 x^{2} + (176) x + 2836 + \frac{45360}{x - 16}$

Simplify the quotient polynomial.

$x^{3} + 11 x^{2} + 176 x + 2836 + \frac{45360}{x - 16}$

Step 11

Since $16 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $16$ is an upper bound for the real roots of the function.

Upper Bound: $16$

Step 12

Apply synthetic division on $\frac{x^{4} - 5 x^{3} + 20 x - 16}{x + 16}$ when $x = - 16$ .

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

$- 16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0				\xa0

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

$- 16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 16)$ and place the result of $(- 16)$ under the next term in the dividend $(- 5)$ .

$- 16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 16$			\xa0
	$1$

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

$- 16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 16$			\xa0
	$1$	$- 21$

Multiply the newest entry in the result $(- 21)$ by the divisor $(- 16)$ and place the result of $(336)$ under the next term in the dividend $(0)$ .

$- 16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 16$	$336$		\xa0
	$1$	$- 21$

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

$- 16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 16$	$336$		\xa0
	$1$	$- 21$	$336$

Multiply the newest entry in the result $(336)$ by the divisor $(- 16)$ and place the result of $(- 5376)$ under the next term in the dividend $(20)$ .

$- 16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 16$	$336$	$- 5376$	\xa0
	$1$	$- 21$	$336$

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

$- 16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 16$	$336$	$- 5376$	\xa0
	$1$	$- 21$	$336$	$- 5356$

Multiply the newest entry in the result $(- 5356)$ by the divisor $(- 16)$ and place the result of $(85696)$ under the next term in the dividend $(- 16)$ .

$- 16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 16$	$336$	$- 5376$	$85696$
	$1$	$- 21$	$336$	$- 5356$

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

$- 16$	$1$	$- 5$	$0$	$20$	$- 16$
	\xa0	$- 16$	$336$	$- 5376$	$85696$
	$1$	$- 21$	$336$	$- 5356$	$85680$

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

$1 x^{3} + - 21 x^{2} + (336) x - 5356 + \frac{85680}{x + 16}$

Simplify the quotient polynomial.

$x^{3} - 21 x^{2} + 336 x - 5356 + \frac{85680}{x + 16}$

Step 13

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 14

Determine the upper and lower bounds.

Upper Bounds: $8, 16$

Lower Bounds: $- 2, - 4, - 8, - 16$

Step 15

Find the Upper and Lower Bounds x^4-5x^3+20x-16

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