Find the Upper and Lower Bounds 6x^3+5x^2-36x-35

$6 x^{3} + 5 x^{2} - 36 x - 35$

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.

$p = \pm 1, \pm 5, \pm 7, \pm 35$

$q = \pm 1, \pm 2, \pm 3, \pm 6$

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{3}, \pm \frac{5}{6}, \pm 7, \pm \frac{7}{2}, \pm \frac{7}{3}, \pm \frac{7}{6}, \pm 35, \pm \frac{35}{2}, \pm \frac{35}{3}, \pm \frac{35}{6}$

Step 2

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x - 5}$ when $x = 5$ .

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

$5$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$5$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(5)$ and place the result of $(30)$ under the next term in the dividend $(5)$ .

$5$	$6$	$5$	$- 36$	$- 35$
	\xa0	$30$		\xa0
	$6$

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

$5$	$6$	$5$	$- 36$	$- 35$
	\xa0	$30$		\xa0
	$6$	$35$

Multiply the newest entry in the result $(35)$ by the divisor $(5)$ and place the result of $(175)$ under the next term in the dividend $(- 36)$ .

$5$	$6$	$5$	$- 36$	$- 35$
	\xa0	$30$	$175$	\xa0
	$6$	$35$

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

$5$	$6$	$5$	$- 36$	$- 35$
	\xa0	$30$	$175$	\xa0
	$6$	$35$	$139$

Multiply the newest entry in the result $(139)$ by the divisor $(5)$ and place the result of $(695)$ under the next term in the dividend $(- 35)$ .

$5$	$6$	$5$	$- 36$	$- 35$
	\xa0	$30$	$175$	$695$
	$6$	$35$	$139$

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

$5$	$6$	$5$	$- 36$	$- 35$
	\xa0	$30$	$175$	$695$
	$6$	$35$	$139$	$660$

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

$6 x^{2} + 35 x + 139 + \frac{660}{x - 5}$

Step 3

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 4

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x + 5}$ when $x = - 5$ .

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

$- 5$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$- 5$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(- 5)$ and place the result of $(- 30)$ under the next term in the dividend $(5)$ .

$- 5$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 30$		\xa0
	$6$

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

$- 5$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 30$		\xa0
	$6$	$- 25$

Multiply the newest entry in the result $(- 25)$ by the divisor $(- 5)$ and place the result of $(125)$ under the next term in the dividend $(- 36)$ .

$- 5$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 30$	$125$	\xa0
	$6$	$- 25$

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

$- 5$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 30$	$125$	\xa0
	$6$	$- 25$	$89$

Multiply the newest entry in the result $(89)$ by the divisor $(- 5)$ and place the result of $(- 445)$ under the next term in the dividend $(- 35)$ .

$- 5$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 30$	$125$	$- 445$
	$6$	$- 25$	$89$

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

$- 5$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 30$	$125$	$- 445$
	$6$	$- 25$	$89$	$- 480$

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

$6 x^{2} + - 25 x + 89 + \frac{- 480}{x + 5}$

Simplify the quotient polynomial.

$6 x^{2} - 25 x + 89 - \frac{480}{x + 5}$

Step 5

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 6

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x - \frac{5}{2}}$ when $x = \frac{5}{2}$ .

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

$\frac{5}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$\frac{5}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(\frac{5}{2})$ and place the result of $(15)$ under the next term in the dividend $(5)$ .

$\frac{5}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$15$		\xa0
	$6$

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

$\frac{5}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$15$		\xa0
	$6$	$20$

Multiply the newest entry in the result $(20)$ by the divisor $(\frac{5}{2})$ and place the result of $(50)$ under the next term in the dividend $(- 36)$ .

$\frac{5}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$15$	$50$	\xa0
	$6$	$20$

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

$\frac{5}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$15$	$50$	\xa0
	$6$	$20$	$14$

Multiply the newest entry in the result $(14)$ by the divisor $(\frac{5}{2})$ and place the result of $(35)$ under the next term in the dividend $(- 35)$ .

$\frac{5}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$15$	$50$	$35$
	$6$	$20$	$14$

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

$\frac{5}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$15$	$50$	$35$
	$6$	$20$	$14$	$0$

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

$6 x^{2} + 20 x + 14$

Step 7

Since $\frac{5}{2} > 0$ and all of the signs in the bottom row of the synthetic division are positive, $\frac{5}{2}$ is an upper bound for the real roots of the function.

Upper Bound: $\frac{5}{2}$

Step 8

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x - 7}$ when $x = 7$ .

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

$7$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$7$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(7)$ and place the result of $(42)$ under the next term in the dividend $(5)$ .

$7$	$6$	$5$	$- 36$	$- 35$
	\xa0	$42$		\xa0
	$6$

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

$7$	$6$	$5$	$- 36$	$- 35$
	\xa0	$42$		\xa0
	$6$	$47$

Multiply the newest entry in the result $(47)$ by the divisor $(7)$ and place the result of $(329)$ under the next term in the dividend $(- 36)$ .

$7$	$6$	$5$	$- 36$	$- 35$
	\xa0	$42$	$329$	\xa0
	$6$	$47$

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

$7$	$6$	$5$	$- 36$	$- 35$
	\xa0	$42$	$329$	\xa0
	$6$	$47$	$293$

Multiply the newest entry in the result $(293)$ by the divisor $(7)$ and place the result of $(2051)$ under the next term in the dividend $(- 35)$ .

$7$	$6$	$5$	$- 36$	$- 35$
	\xa0	$42$	$329$	$2051$
	$6$	$47$	$293$

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

$7$	$6$	$5$	$- 36$	$- 35$
	\xa0	$42$	$329$	$2051$
	$6$	$47$	$293$	$2016$

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

$6 x^{2} + 47 x + 293 + \frac{2016}{x - 7}$

Step 9

Since $7 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $7$ is an upper bound for the real roots of the function.

Upper Bound: $7$

Step 10

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x + 7}$ when $x = - 7$ .

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

$- 7$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$- 7$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(- 7)$ and place the result of $(- 42)$ under the next term in the dividend $(5)$ .

$- 7$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 42$		\xa0
	$6$

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

$- 7$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 42$		\xa0
	$6$	$- 37$

Multiply the newest entry in the result $(- 37)$ by the divisor $(- 7)$ and place the result of $(259)$ under the next term in the dividend $(- 36)$ .

$- 7$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 42$	$259$	\xa0
	$6$	$- 37$

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

$- 7$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 42$	$259$	\xa0
	$6$	$- 37$	$223$

Multiply the newest entry in the result $(223)$ by the divisor $(- 7)$ and place the result of $(- 1561)$ under the next term in the dividend $(- 35)$ .

$- 7$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 42$	$259$	$- 1561$
	$6$	$- 37$	$223$

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

$- 7$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 42$	$259$	$- 1561$
	$6$	$- 37$	$223$	$- 1596$

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

$6 x^{2} + - 37 x + 223 + \frac{- 1596}{x + 7}$

Simplify the quotient polynomial.

$6 x^{2} - 37 x + 223 - \frac{1596}{x + 7}$

Step 11

Since $- 7 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 7$ is a lower bound for the real roots of the function.

Lower Bound: $- 7$

Step 12

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x - \frac{7}{2}}$ when $x = \frac{7}{2}$ .

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

$\frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$\frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(\frac{7}{2})$ and place the result of $(21)$ under the next term in the dividend $(5)$ .

$\frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$21$		\xa0
	$6$

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

$\frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$21$		\xa0
	$6$	$26$

Multiply the newest entry in the result $(26)$ by the divisor $(\frac{7}{2})$ and place the result of $(91)$ under the next term in the dividend $(- 36)$ .

$\frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$21$	$91$	\xa0
	$6$	$26$

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

$\frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$21$	$91$	\xa0
	$6$	$26$	$55$

Multiply the newest entry in the result $(55)$ by the divisor $(\frac{7}{2})$ and place the result of $(\frac{385}{2})$ under the next term in the dividend $(- 35)$ .

$\frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$21$	$91$	$\frac{385}{2}$
	$6$	$26$	$55$

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

$\frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$21$	$91$	$\frac{385}{2}$
	$6$	$26$	$55$	$\frac{315}{2}$

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

$6 x^{2} + 26 x + 55 + \frac{\frac{315}{2}}{x - \frac{7}{2}}$

Simplify the quotient polynomial.

$6 x^{2} + 26 x + 55 + \frac{315}{2 x - 7}$

Step 13

Since $\frac{7}{2} > 0$ and all of the signs in the bottom row of the synthetic division are positive, $\frac{7}{2}$ is an upper bound for the real roots of the function.

Upper Bound: $\frac{7}{2}$

Step 14

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x + \frac{7}{2}}$ when $x = - \frac{7}{2}$ .

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

$- \frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$- \frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(- \frac{7}{2})$ and place the result of $(- 21)$ under the next term in the dividend $(5)$ .

$- \frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 21$		\xa0
	$6$

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

$- \frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 21$		\xa0
	$6$	$- 16$

Multiply the newest entry in the result $(- 16)$ by the divisor $(- \frac{7}{2})$ and place the result of $(56)$ under the next term in the dividend $(- 36)$ .

$- \frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 21$	$56$	\xa0
	$6$	$- 16$

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

$- \frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 21$	$56$	\xa0
	$6$	$- 16$	$20$

Multiply the newest entry in the result $(20)$ by the divisor $(- \frac{7}{2})$ and place the result of $(- 70)$ under the next term in the dividend $(- 35)$ .

$- \frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 21$	$56$	$- 70$
	$6$	$- 16$	$20$

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

$- \frac{7}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 21$	$56$	$- 70$
	$6$	$- 16$	$20$	$- 105$

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

$6 x^{2} + - 16 x + 20 + \frac{- 105}{x + \frac{7}{2}}$

Simplify the quotient polynomial.

$6 x^{2} - 16 x + 20 - \frac{210}{2 x + 7}$

Step 15

Since $- \frac{7}{2} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{7}{2}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{7}{2}$

Step 16

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x - 35}$ when $x = 35$ .

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

$35$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$35$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(35)$ and place the result of $(210)$ under the next term in the dividend $(5)$ .

$35$	$6$	$5$	$- 36$	$- 35$
	\xa0	$210$		\xa0
	$6$

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

$35$	$6$	$5$	$- 36$	$- 35$
	\xa0	$210$		\xa0
	$6$	$215$

Multiply the newest entry in the result $(215)$ by the divisor $(35)$ and place the result of $(7525)$ under the next term in the dividend $(- 36)$ .

$35$	$6$	$5$	$- 36$	$- 35$
	\xa0	$210$	$7525$	\xa0
	$6$	$215$

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

$35$	$6$	$5$	$- 36$	$- 35$
	\xa0	$210$	$7525$	\xa0
	$6$	$215$	$7489$

Multiply the newest entry in the result $(7489)$ by the divisor $(35)$ and place the result of $(262115)$ under the next term in the dividend $(- 35)$ .

$35$	$6$	$5$	$- 36$	$- 35$
	\xa0	$210$	$7525$	$262115$
	$6$	$215$	$7489$

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

$35$	$6$	$5$	$- 36$	$- 35$
	\xa0	$210$	$7525$	$262115$
	$6$	$215$	$7489$	$262080$

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

$6 x^{2} + 215 x + 7489 + \frac{262080}{x - 35}$

Step 17

Since $35 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $35$ is an upper bound for the real roots of the function.

Upper Bound: $35$

Step 18

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x + 35}$ when $x = - 35$ .

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

$- 35$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$- 35$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(- 35)$ and place the result of $(- 210)$ under the next term in the dividend $(5)$ .

$- 35$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 210$		\xa0
	$6$

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

$- 35$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 210$		\xa0
	$6$	$- 205$

Multiply the newest entry in the result $(- 205)$ by the divisor $(- 35)$ and place the result of $(7175)$ under the next term in the dividend $(- 36)$ .

$- 35$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 210$	$7175$	\xa0
	$6$	$- 205$

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

$- 35$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 210$	$7175$	\xa0
	$6$	$- 205$	$7139$

Multiply the newest entry in the result $(7139)$ by the divisor $(- 35)$ and place the result of $(- 249865)$ under the next term in the dividend $(- 35)$ .

$- 35$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 210$	$7175$	$- 249865$
	$6$	$- 205$	$7139$

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

$- 35$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 210$	$7175$	$- 249865$
	$6$	$- 205$	$7139$	$- 249900$

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

$6 x^{2} + - 205 x + 7139 + \frac{- 249900}{x + 35}$

Simplify the quotient polynomial.

$6 x^{2} - 205 x + 7139 - \frac{249900}{x + 35}$

Step 19

Since $- 35 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 35$ is a lower bound for the real roots of the function.

Lower Bound: $- 35$

Step 20

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x - \frac{35}{2}}$ when $x = \frac{35}{2}$ .

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

$\frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$\frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(\frac{35}{2})$ and place the result of $(105)$ under the next term in the dividend $(5)$ .

$\frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$105$		\xa0
	$6$

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

$\frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$105$		\xa0
	$6$	$110$

Multiply the newest entry in the result $(110)$ by the divisor $(\frac{35}{2})$ and place the result of $(1925)$ under the next term in the dividend $(- 36)$ .

$\frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$105$	$1925$	\xa0
	$6$	$110$

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

$\frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$105$	$1925$	\xa0
	$6$	$110$	$1889$

Multiply the newest entry in the result $(1889)$ by the divisor $(\frac{35}{2})$ and place the result of $(\frac{66115}{2})$ under the next term in the dividend $(- 35)$ .

$\frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$105$	$1925$	$\frac{66115}{2}$
	$6$	$110$	$1889$

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

$\frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$105$	$1925$	$\frac{66115}{2}$
	$6$	$110$	$1889$	$\frac{66045}{2}$

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

$6 x^{2} + 110 x + 1889 + \frac{\frac{66045}{2}}{x - \frac{35}{2}}$

Simplify the quotient polynomial.

$6 x^{2} + 110 x + 1889 + \frac{66045}{2 x - 35}$

Step 21

Since $\frac{35}{2} > 0$ and all of the signs in the bottom row of the synthetic division are positive, $\frac{35}{2}$ is an upper bound for the real roots of the function.

Upper Bound: $\frac{35}{2}$

Step 22

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x + \frac{35}{2}}$ when $x = - \frac{35}{2}$ .

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

$- \frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$- \frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(- \frac{35}{2})$ and place the result of $(- 105)$ under the next term in the dividend $(5)$ .

$- \frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 105$		\xa0
	$6$

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

$- \frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 105$		\xa0
	$6$	$- 100$

Multiply the newest entry in the result $(- 100)$ by the divisor $(- \frac{35}{2})$ and place the result of $(1750)$ under the next term in the dividend $(- 36)$ .

$- \frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 105$	$1750$	\xa0
	$6$	$- 100$

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

$- \frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 105$	$1750$	\xa0
	$6$	$- 100$	$1714$

Multiply the newest entry in the result $(1714)$ by the divisor $(- \frac{35}{2})$ and place the result of $(- 29995)$ under the next term in the dividend $(- 35)$ .

$- \frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 105$	$1750$	$- 29995$
	$6$	$- 100$	$1714$

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

$- \frac{35}{2}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 105$	$1750$	$- 29995$
	$6$	$- 100$	$1714$	$- 30030$

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

$6 x^{2} + - 100 x + 1714 + \frac{- 30030}{x + \frac{35}{2}}$

Simplify the quotient polynomial.

$6 x^{2} - 100 x + 1714 - \frac{60060}{2 x + 35}$

Step 23

Since $- \frac{35}{2} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{35}{2}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{35}{2}$

Step 24

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x - \frac{35}{3}}$ when $x = \frac{35}{3}$ .

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

$\frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$\frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(\frac{35}{3})$ and place the result of $(70)$ under the next term in the dividend $(5)$ .

$\frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$70$		\xa0
	$6$

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

$\frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$70$		\xa0
	$6$	$75$

Multiply the newest entry in the result $(75)$ by the divisor $(\frac{35}{3})$ and place the result of $(875)$ under the next term in the dividend $(- 36)$ .

$\frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$70$	$875$	\xa0
	$6$	$75$

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

$\frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$70$	$875$	\xa0
	$6$	$75$	$839$

Multiply the newest entry in the result $(839)$ by the divisor $(\frac{35}{3})$ and place the result of $(\frac{29365}{3})$ under the next term in the dividend $(- 35)$ .

$\frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$70$	$875$	$\frac{29365}{3}$
	$6$	$75$	$839$

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

$\frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$70$	$875$	$\frac{29365}{3}$
	$6$	$75$	$839$	$\frac{29260}{3}$

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

$6 x^{2} + 75 x + 839 + \frac{\frac{29260}{3}}{x - \frac{35}{3}}$

Simplify the quotient polynomial.

$6 x^{2} + 75 x + 839 + \frac{29260}{3 x - 35}$

Step 25

Since $\frac{35}{3} > 0$ and all of the signs in the bottom row of the synthetic division are positive, $\frac{35}{3}$ is an upper bound for the real roots of the function.

Upper Bound: $\frac{35}{3}$

Step 26

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x + \frac{35}{3}}$ when $x = - \frac{35}{3}$ .

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

$- \frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$- \frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(- \frac{35}{3})$ and place the result of $(- 70)$ under the next term in the dividend $(5)$ .

$- \frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 70$		\xa0
	$6$

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

$- \frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 70$		\xa0
	$6$	$- 65$

Multiply the newest entry in the result $(- 65)$ by the divisor $(- \frac{35}{3})$ and place the result of $(\frac{2275}{3})$ under the next term in the dividend $(- 36)$ .

$- \frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 70$	$\frac{2275}{3}$	\xa0
	$6$	$- 65$

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

$- \frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 70$	$\frac{2275}{3}$	\xa0
	$6$	$- 65$	$\frac{2167}{3}$

Multiply the newest entry in the result $(\frac{2167}{3})$ by the divisor $(- \frac{35}{3})$ and place the result of $(- \frac{75845}{9})$ under the next term in the dividend $(- 35)$ .

$- \frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 70$	$\frac{2275}{3}$	$- \frac{75845}{9}$
	$6$	$- 65$	$\frac{2167}{3}$

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

$- \frac{35}{3}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 70$	$\frac{2275}{3}$	$- \frac{75845}{9}$
	$6$	$- 65$	$\frac{2167}{3}$	$- \frac{76160}{9}$

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

$6 x^{2} + - 65 x + \frac{2167}{3} + \frac{- \frac{76160}{9}}{x + \frac{35}{3}}$

Simplify the quotient polynomial.

$6 x^{2} - 65 x + \frac{2167}{3} - \frac{76160}{3 (3 x + 35)}$

Step 27

Since $- \frac{35}{3} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{35}{3}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{35}{3}$

Step 28

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x - \frac{35}{6}}$ when $x = \frac{35}{6}$ .

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

$\frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$\frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(\frac{35}{6})$ and place the result of $(35)$ under the next term in the dividend $(5)$ .

$\frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$35$		\xa0
	$6$

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

$\frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$35$		\xa0
	$6$	$40$

Multiply the newest entry in the result $(40)$ by the divisor $(\frac{35}{6})$ and place the result of $(\frac{700}{3})$ under the next term in the dividend $(- 36)$ .

$\frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$35$	$\frac{700}{3}$	\xa0
	$6$	$40$

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

$\frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$35$	$\frac{700}{3}$	\xa0
	$6$	$40$	$\frac{592}{3}$

Multiply the newest entry in the result $(\frac{592}{3})$ by the divisor $(\frac{35}{6})$ and place the result of $(\frac{10360}{9})$ under the next term in the dividend $(- 35)$ .

$\frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$35$	$\frac{700}{3}$	$\frac{10360}{9}$
	$6$	$40$	$\frac{592}{3}$

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

$\frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$35$	$\frac{700}{3}$	$\frac{10360}{9}$
	$6$	$40$	$\frac{592}{3}$	$\frac{10045}{9}$

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

$6 x^{2} + 40 x + \frac{592}{3} + \frac{\frac{10045}{9}}{x - \frac{35}{6}}$

Simplify the quotient polynomial.

$6 x^{2} + 40 x + \frac{592}{3} + \frac{20090}{3 (6 x - 35)}$

Step 29

Since $\frac{35}{6} > 0$ and all of the signs in the bottom row of the synthetic division are positive, $\frac{35}{6}$ is an upper bound for the real roots of the function.

Upper Bound: $\frac{35}{6}$

Step 30

Apply synthetic division on $\frac{6 x^{3} + 5 x^{2} - 36 x - 35}{x + \frac{35}{6}}$ when $x = - \frac{35}{6}$ .

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

$- \frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$- \frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(- \frac{35}{6})$ and place the result of $(- 35)$ under the next term in the dividend $(5)$ .

$- \frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 35$		\xa0
	$6$

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

$- \frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 35$		\xa0
	$6$	$- 30$

Multiply the newest entry in the result $(- 30)$ by the divisor $(- \frac{35}{6})$ and place the result of $(175)$ under the next term in the dividend $(- 36)$ .

$- \frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 35$	$175$	\xa0
	$6$	$- 30$

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

$- \frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 35$	$175$	\xa0
	$6$	$- 30$	$139$

Multiply the newest entry in the result $(139)$ by the divisor $(- \frac{35}{6})$ and place the result of $(- \frac{4865}{6})$ under the next term in the dividend $(- 35)$ .

$- \frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 35$	$175$	$- \frac{4865}{6}$
	$6$	$- 30$	$139$

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

$- \frac{35}{6}$	$6$	$5$	$- 36$	$- 35$
	\xa0	$- 35$	$175$	$- \frac{4865}{6}$
	$6$	$- 30$	$139$	$- \frac{5075}{6}$

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

$6 x^{2} + - 30 x + 139 + \frac{- \frac{5075}{6}}{x + \frac{35}{6}}$

Simplify the quotient polynomial.

$6 x^{2} - 30 x + 139 - \frac{5075}{6 x + 35}$

Step 31

Since $- \frac{35}{6} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{35}{6}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{35}{6}$

Step 32

Determine the upper and lower bounds.

Upper Bounds: $5, \frac{5}{2}, 7, \frac{7}{2}, 35, \frac{35}{2}, \frac{35}{3}, \frac{35}{6}$

Lower Bounds: $- 5, - 7, - \frac{7}{2}, - 35, - \frac{35}{2}, - \frac{35}{3}, - \frac{35}{6}$

Step 33

Find the Upper and Lower Bounds 6x^3+5x^2-36x-35

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