Find the Upper and Lower Bounds x^4+5x^3+2x^2-20x-24

$x^{4} + 5 x^{3} + 2 x^{2} - 20 x - 24$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$

$q = \pm 1$

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

$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$

Step 2

Apply synthetic division on $\frac{x^{4} + 5 x^{3} + 2 x^{2} - 20 x - 24}{x - 2}$ when $x = 2$ .

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

$2$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0

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

$2$	$1$	$5$	$2$	$- 20$	$- 24$
	\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 $(5)$ .

$2$	$1$	$5$	$2$	$- 20$	$- 24$
	\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.

$2$	$1$	$5$	$2$	$- 20$	$- 24$
	\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 $(2)$ .

$2$	$1$	$5$	$2$	$- 20$	$- 24$
	\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.

$2$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$2$	$14$		\xa0
	$1$	$7$	$16$

Multiply the newest entry in the result $(16)$ by the divisor $(2)$ and place the result of $(32)$ under the next term in the dividend $(- 20)$ .

$2$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$2$	$14$	$32$	\xa0
	$1$	$7$	$16$

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

$2$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$2$	$14$	$32$	\xa0
	$1$	$7$	$16$	$12$

Multiply the newest entry in the result $(12)$ by the divisor $(2)$ and place the result of $(24)$ under the next term in the dividend $(- 24)$ .

$2$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$2$	$14$	$32$	$24$
	$1$	$7$	$16$	$12$

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

$2$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$2$	$14$	$32$	$24$
	$1$	$7$	$16$	$12$	$0$

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

$1 x^{3} + 7 x^{2} + (16) x + 12$

Simplify the quotient polynomial.

$x^{3} + 7 x^{2} + 16 x + 12$

Step 3

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

Upper Bound: $2$

Step 4

Apply synthetic division on $\frac{x^{4} + 5 x^{3} + 2 x^{2} - 20 x - 24}{x - 3}$ when $x = 3$ .

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

$3$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0

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

$3$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(3)$ and place the result of $(3)$ under the next term in the dividend $(5)$ .

$3$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$3$			\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.

$3$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$3$			\xa0
	$1$	$8$

Multiply the newest entry in the result $(8)$ by the divisor $(3)$ and place the result of $(24)$ under the next term in the dividend $(2)$ .

$3$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$3$	$24$		\xa0
	$1$	$8$

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

$3$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$3$	$24$		\xa0
	$1$	$8$	$26$

Multiply the newest entry in the result $(26)$ by the divisor $(3)$ and place the result of $(78)$ under the next term in the dividend $(- 20)$ .

$3$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$3$	$24$	$78$	\xa0
	$1$	$8$	$26$

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

$3$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$3$	$24$	$78$	\xa0
	$1$	$8$	$26$	$58$

Multiply the newest entry in the result $(58)$ by the divisor $(3)$ and place the result of $(174)$ under the next term in the dividend $(- 24)$ .

$3$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$3$	$24$	$78$	$174$
	$1$	$8$	$26$	$58$

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

$3$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$3$	$24$	$78$	$174$
	$1$	$8$	$26$	$58$	$150$

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

$1 x^{3} + 8 x^{2} + (26) x + 58 + \frac{150}{x - 3}$

Simplify the quotient polynomial.

$x^{3} + 8 x^{2} + 26 x + 58 + \frac{150}{x - 3}$

Step 5

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 6

Apply synthetic division on $\frac{x^{4} + 5 x^{3} + 2 x^{2} - 20 x - 24}{x - 4}$ when $x = 4$ .

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

$4$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0

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

$4$	$1$	$5$	$2$	$- 20$	$- 24$
	\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 $(5)$ .

$4$	$1$	$5$	$2$	$- 20$	$- 24$
	\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.

$4$	$1$	$5$	$2$	$- 20$	$- 24$
	\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 $(2)$ .

$4$	$1$	$5$	$2$	$- 20$	$- 24$
	\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.

$4$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$4$	$36$		\xa0
	$1$	$9$	$38$

Multiply the newest entry in the result $(38)$ by the divisor $(4)$ and place the result of $(152)$ under the next term in the dividend $(- 20)$ .

$4$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$4$	$36$	$152$	\xa0
	$1$	$9$	$38$

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

$4$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$4$	$36$	$152$	\xa0
	$1$	$9$	$38$	$132$

Multiply the newest entry in the result $(132)$ by the divisor $(4)$ and place the result of $(528)$ under the next term in the dividend $(- 24)$ .

$4$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$4$	$36$	$152$	$528$
	$1$	$9$	$38$	$132$

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

$4$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$4$	$36$	$152$	$528$
	$1$	$9$	$38$	$132$	$504$

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

$1 x^{3} + 9 x^{2} + (38) x + 132 + \frac{504}{x - 4}$

Simplify the quotient polynomial.

$x^{3} + 9 x^{2} + 38 x + 132 + \frac{504}{x - 4}$

Step 7

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

Upper Bound: $4$

Step 8

Apply synthetic division on $\frac{x^{4} + 5 x^{3} + 2 x^{2} - 20 x - 24}{x - 6}$ when $x = 6$ .

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

$6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0

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

$6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(6)$ and place the result of $(6)$ under the next term in the dividend $(5)$ .

$6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$6$			\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.

$6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$6$			\xa0
	$1$	$11$

Multiply the newest entry in the result $(11)$ by the divisor $(6)$ and place the result of $(66)$ under the next term in the dividend $(2)$ .

$6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$6$	$66$		\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.

$6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$6$	$66$		\xa0
	$1$	$11$	$68$

Multiply the newest entry in the result $(68)$ by the divisor $(6)$ and place the result of $(408)$ under the next term in the dividend $(- 20)$ .

$6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$6$	$66$	$408$	\xa0
	$1$	$11$	$68$

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

$6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$6$	$66$	$408$	\xa0
	$1$	$11$	$68$	$388$

Multiply the newest entry in the result $(388)$ by the divisor $(6)$ and place the result of $(2328)$ under the next term in the dividend $(- 24)$ .

$6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$6$	$66$	$408$	$2328$
	$1$	$11$	$68$	$388$

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

$6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$6$	$66$	$408$	$2328$
	$1$	$11$	$68$	$388$	$2304$

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

$1 x^{3} + 11 x^{2} + (68) x + 388 + \frac{2304}{x - 6}$

Simplify the quotient polynomial.

$x^{3} + 11 x^{2} + 68 x + 388 + \frac{2304}{x - 6}$

Step 9

Since $6 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $6$ is an upper bound for the real roots of the function.

Upper Bound: $6$

Step 10

Apply synthetic division on $\frac{x^{4} + 5 x^{3} + 2 x^{2} - 20 x - 24}{x + 6}$ when $x = - 6$ .

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

$- 6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0

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

$- 6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 6)$ and place the result of $(- 6)$ under the next term in the dividend $(5)$ .

$- 6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 6$			\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.

$- 6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 6$			\xa0
	$1$	$- 1$

Multiply the newest entry in the result $(- 1)$ by the divisor $(- 6)$ and place the result of $(6)$ under the next term in the dividend $(2)$ .

$- 6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 6$	$6$		\xa0
	$1$	$- 1$

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

$- 6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 6$	$6$		\xa0
	$1$	$- 1$	$8$

Multiply the newest entry in the result $(8)$ by the divisor $(- 6)$ and place the result of $(- 48)$ under the next term in the dividend $(- 20)$ .

$- 6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 6$	$6$	$- 48$	\xa0
	$1$	$- 1$	$8$

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

$- 6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 6$	$6$	$- 48$	\xa0
	$1$	$- 1$	$8$	$- 68$

Multiply the newest entry in the result $(- 68)$ by the divisor $(- 6)$ and place the result of $(408)$ under the next term in the dividend $(- 24)$ .

$- 6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 6$	$6$	$- 48$	$408$
	$1$	$- 1$	$8$	$- 68$

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

$- 6$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 6$	$6$	$- 48$	$408$
	$1$	$- 1$	$8$	$- 68$	$384$

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

$1 x^{3} + - 1 x^{2} + (8) x - 68 + \frac{384}{x + 6}$

Simplify the quotient polynomial.

$x^{3} - x^{2} + 8 x - 68 + \frac{384}{x + 6}$

Step 11

Since $- 6 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 6$ is a lower bound for the real roots of the function.

Lower Bound: $- 6$

Step 12

Apply synthetic division on $\frac{x^{4} + 5 x^{3} + 2 x^{2} - 20 x - 24}{x - 8}$ when $x = 8$ .

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

$8$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0

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

$8$	$1$	$5$	$2$	$- 20$	$- 24$
	\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 $(5)$ .

$8$	$1$	$5$	$2$	$- 20$	$- 24$
	\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.

$8$	$1$	$5$	$2$	$- 20$	$- 24$
	\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 $(2)$ .

$8$	$1$	$5$	$2$	$- 20$	$- 24$
	\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.

$8$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$8$	$104$		\xa0
	$1$	$13$	$106$

Multiply the newest entry in the result $(106)$ by the divisor $(8)$ and place the result of $(848)$ under the next term in the dividend $(- 20)$ .

$8$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$8$	$104$	$848$	\xa0
	$1$	$13$	$106$

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

$8$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$8$	$104$	$848$	\xa0
	$1$	$13$	$106$	$828$

Multiply the newest entry in the result $(828)$ by the divisor $(8)$ and place the result of $(6624)$ under the next term in the dividend $(- 24)$ .

$8$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$8$	$104$	$848$	$6624$
	$1$	$13$	$106$	$828$

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

$8$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$8$	$104$	$848$	$6624$
	$1$	$13$	$106$	$828$	$6600$

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

$1 x^{3} + 13 x^{2} + (106) x + 828 + \frac{6600}{x - 8}$

Simplify the quotient polynomial.

$x^{3} + 13 x^{2} + 106 x + 828 + \frac{6600}{x - 8}$

Step 13

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 14

Apply synthetic division on $\frac{x^{4} + 5 x^{3} + 2 x^{2} - 20 x - 24}{x + 8}$ when $x = - 8$ .

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

$- 8$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0

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

$- 8$	$1$	$5$	$2$	$- 20$	$- 24$
	\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 $(5)$ .

$- 8$	$1$	$5$	$2$	$- 20$	$- 24$
	\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.

$- 8$	$1$	$5$	$2$	$- 20$	$- 24$
	\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 $(2)$ .

$- 8$	$1$	$5$	$2$	$- 20$	$- 24$
	\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.

$- 8$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 8$	$24$		\xa0
	$1$	$- 3$	$26$

Multiply the newest entry in the result $(26)$ by the divisor $(- 8)$ and place the result of $(- 208)$ under the next term in the dividend $(- 20)$ .

$- 8$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 8$	$24$	$- 208$	\xa0
	$1$	$- 3$	$26$

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

$- 8$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 8$	$24$	$- 208$	\xa0
	$1$	$- 3$	$26$	$- 228$

Multiply the newest entry in the result $(- 228)$ by the divisor $(- 8)$ and place the result of $(1824)$ under the next term in the dividend $(- 24)$ .

$- 8$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 8$	$24$	$- 208$	$1824$
	$1$	$- 3$	$26$	$- 228$

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

$- 8$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 8$	$24$	$- 208$	$1824$
	$1$	$- 3$	$26$	$- 228$	$1800$

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

$1 x^{3} + - 3 x^{2} + (26) x - 228 + \frac{1800}{x + 8}$

Simplify the quotient polynomial.

$x^{3} - 3 x^{2} + 26 x - 228 + \frac{1800}{x + 8}$

Step 15

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 16

Apply synthetic division on $\frac{x^{4} + 5 x^{3} + 2 x^{2} - 20 x - 24}{x - 12}$ when $x = 12$ .

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

$12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0

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

$12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(12)$ and place the result of $(12)$ under the next term in the dividend $(5)$ .

$12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$12$			\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.

$12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$12$			\xa0
	$1$	$17$

Multiply the newest entry in the result $(17)$ by the divisor $(12)$ and place the result of $(204)$ under the next term in the dividend $(2)$ .

$12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$12$	$204$		\xa0
	$1$	$17$

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

$12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$12$	$204$		\xa0
	$1$	$17$	$206$

Multiply the newest entry in the result $(206)$ by the divisor $(12)$ and place the result of $(2472)$ under the next term in the dividend $(- 20)$ .

$12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$12$	$204$	$2472$	\xa0
	$1$	$17$	$206$

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

$12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$12$	$204$	$2472$	\xa0
	$1$	$17$	$206$	$2452$

Multiply the newest entry in the result $(2452)$ by the divisor $(12)$ and place the result of $(29424)$ under the next term in the dividend $(- 24)$ .

$12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$12$	$204$	$2472$	$29424$
	$1$	$17$	$206$	$2452$

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

$12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$12$	$204$	$2472$	$29424$
	$1$	$17$	$206$	$2452$	$29400$

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

$1 x^{3} + 17 x^{2} + (206) x + 2452 + \frac{29400}{x - 12}$

Simplify the quotient polynomial.

$x^{3} + 17 x^{2} + 206 x + 2452 + \frac{29400}{x - 12}$

Step 17

Since $12 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $12$ is an upper bound for the real roots of the function.

Upper Bound: $12$

Step 18

Apply synthetic division on $\frac{x^{4} + 5 x^{3} + 2 x^{2} - 20 x - 24}{x + 12}$ when $x = - 12$ .

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

$- 12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0

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

$- 12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 12)$ and place the result of $(- 12)$ under the next term in the dividend $(5)$ .

$- 12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 12$			\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.

$- 12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 12$			\xa0
	$1$	$- 7$

Multiply the newest entry in the result $(- 7)$ by the divisor $(- 12)$ and place the result of $(84)$ under the next term in the dividend $(2)$ .

$- 12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 12$	$84$		\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.

$- 12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 12$	$84$		\xa0
	$1$	$- 7$	$86$

Multiply the newest entry in the result $(86)$ by the divisor $(- 12)$ and place the result of $(- 1032)$ under the next term in the dividend $(- 20)$ .

$- 12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 12$	$84$	$- 1032$	\xa0
	$1$	$- 7$	$86$

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

$- 12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 12$	$84$	$- 1032$	\xa0
	$1$	$- 7$	$86$	$- 1052$

Multiply the newest entry in the result $(- 1052)$ by the divisor $(- 12)$ and place the result of $(12624)$ under the next term in the dividend $(- 24)$ .

$- 12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 12$	$84$	$- 1032$	$12624$
	$1$	$- 7$	$86$	$- 1052$

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

$- 12$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 12$	$84$	$- 1032$	$12624$
	$1$	$- 7$	$86$	$- 1052$	$12600$

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

$1 x^{3} + - 7 x^{2} + (86) x - 1052 + \frac{12600}{x + 12}$

Simplify the quotient polynomial.

$x^{3} - 7 x^{2} + 86 x - 1052 + \frac{12600}{x + 12}$

Step 19

Since $- 12 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 12$ is a lower bound for the real roots of the function.

Lower Bound: $- 12$

Step 20

Apply synthetic division on $\frac{x^{4} + 5 x^{3} + 2 x^{2} - 20 x - 24}{x - 24}$ when $x = 24$ .

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

$24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0

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

$24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(24)$ and place the result of $(24)$ under the next term in the dividend $(5)$ .

$24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$24$			\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.

$24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$24$			\xa0
	$1$	$29$

Multiply the newest entry in the result $(29)$ by the divisor $(24)$ and place the result of $(696)$ under the next term in the dividend $(2)$ .

$24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$24$	$696$		\xa0
	$1$	$29$

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

$24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$24$	$696$		\xa0
	$1$	$29$	$698$

Multiply the newest entry in the result $(698)$ by the divisor $(24)$ and place the result of $(16752)$ under the next term in the dividend $(- 20)$ .

$24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$24$	$696$	$16752$	\xa0
	$1$	$29$	$698$

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

$24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$24$	$696$	$16752$	\xa0
	$1$	$29$	$698$	$16732$

Multiply the newest entry in the result $(16732)$ by the divisor $(24)$ and place the result of $(401568)$ under the next term in the dividend $(- 24)$ .

$24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$24$	$696$	$16752$	$401568$
	$1$	$29$	$698$	$16732$

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

$24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$24$	$696$	$16752$	$401568$
	$1$	$29$	$698$	$16732$	$401544$

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

$1 x^{3} + 29 x^{2} + (698) x + 16732 + \frac{401544}{x - 24}$

Simplify the quotient polynomial.

$x^{3} + 29 x^{2} + 698 x + 16732 + \frac{401544}{x - 24}$

Step 21

Since $24 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $24$ is an upper bound for the real roots of the function.

Upper Bound: $24$

Step 22

Apply synthetic division on $\frac{x^{4} + 5 x^{3} + 2 x^{2} - 20 x - 24}{x + 24}$ when $x = - 24$ .

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

$- 24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0

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

$- 24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 24)$ and place the result of $(- 24)$ under the next term in the dividend $(5)$ .

$- 24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 24$			\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.

$- 24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 24$			\xa0
	$1$	$- 19$

Multiply the newest entry in the result $(- 19)$ by the divisor $(- 24)$ and place the result of $(456)$ under the next term in the dividend $(2)$ .

$- 24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 24$	$456$		\xa0
	$1$	$- 19$

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

$- 24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 24$	$456$		\xa0
	$1$	$- 19$	$458$

Multiply the newest entry in the result $(458)$ by the divisor $(- 24)$ and place the result of $(- 10992)$ under the next term in the dividend $(- 20)$ .

$- 24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 24$	$456$	$- 10992$	\xa0
	$1$	$- 19$	$458$

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

$- 24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 24$	$456$	$- 10992$	\xa0
	$1$	$- 19$	$458$	$- 11012$

Multiply the newest entry in the result $(- 11012)$ by the divisor $(- 24)$ and place the result of $(264288)$ under the next term in the dividend $(- 24)$ .

$- 24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 24$	$456$	$- 10992$	$264288$
	$1$	$- 19$	$458$	$- 11012$

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

$- 24$	$1$	$5$	$2$	$- 20$	$- 24$
	\xa0	$- 24$	$456$	$- 10992$	$264288$
	$1$	$- 19$	$458$	$- 11012$	$264264$

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

$1 x^{3} + - 19 x^{2} + (458) x - 11012 + \frac{264264}{x + 24}$

Simplify the quotient polynomial.

$x^{3} - 19 x^{2} + 458 x - 11012 + \frac{264264}{x + 24}$

Step 23

Since $- 24 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 24$ is a lower bound for the real roots of the function.

Lower Bound: $- 24$

Step 24

Determine the upper and lower bounds.

Upper Bounds: $2, 3, 4, 6, 8, 12, 24$

Lower Bounds: $- 6, - 8, - 12, - 24$

Step 25

Find the Upper and Lower Bounds x^4+5x^3+2x^2-20x-24

Related Questions

Support LCMGCF.com by disabling your adblocker.