Find the Upper and Lower Bounds 3x^3-4x^2-x+6

$3 x^{3} - 4 x^{2} - x + 6$

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.

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

$q = \pm 1, \pm 3$

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

$\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm 6$

Step 2

Apply synthetic division on $\frac{3 x^{3} - 4 x^{2} - x + 6}{x + 1}$ when $x = - 1$ .

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

$- 1$	$3$	$- 4$	$- 1$	$6$
	\xa0			\xa0

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

$- 1$	$3$	$- 4$	$- 1$	$6$
	\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 $(- 4)$ .

$- 1$	$3$	$- 4$	$- 1$	$6$
	\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.

$- 1$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 3$		\xa0
	$3$	$- 7$

Multiply the newest entry in the result $(- 7)$ by the divisor $(- 1)$ and place the result of $(7)$ under the next term in the dividend $(- 1)$ .

$- 1$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 3$	$7$	\xa0
	$3$	$- 7$

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

$- 1$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 3$	$7$	\xa0
	$3$	$- 7$	$6$

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

$- 1$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 3$	$7$	$- 6$
	$3$	$- 7$	$6$

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

$- 1$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 3$	$7$	$- 6$
	$3$	$- 7$	$6$	$0$

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

$3 x^{2} + - 7 x + 6$

Simplify the quotient polynomial.

$3 x^{2} - 7 x + 6$

Step 3

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 4

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

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

$2$	$3$	$- 4$	$- 1$	$6$
	\xa0			\xa0

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

$2$	$3$	$- 4$	$- 1$	$6$
	\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 $(- 4)$ .

$2$	$3$	$- 4$	$- 1$	$6$
	\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.

$2$	$3$	$- 4$	$- 1$	$6$
	\xa0	$6$		\xa0
	$3$	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(2)$ and place the result of $(4)$ under the next term in the dividend $(- 1)$ .

$2$	$3$	$- 4$	$- 1$	$6$
	\xa0	$6$	$4$	\xa0
	$3$	$2$

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

$2$	$3$	$- 4$	$- 1$	$6$
	\xa0	$6$	$4$	\xa0
	$3$	$2$	$3$

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 $(6)$ .

$2$	$3$	$- 4$	$- 1$	$6$
	\xa0	$6$	$4$	$6$
	$3$	$2$	$3$

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

$2$	$3$	$- 4$	$- 1$	$6$
	\xa0	$6$	$4$	$6$
	$3$	$2$	$3$	$12$

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

$3 x^{2} + 2 x + 3 + \frac{12}{x - 2}$

Step 5

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 6

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

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

$- 2$	$3$	$- 4$	$- 1$	$6$
	\xa0			\xa0

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

$- 2$	$3$	$- 4$	$- 1$	$6$
	\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 $(- 4)$ .

$- 2$	$3$	$- 4$	$- 1$	$6$
	\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.

$- 2$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 6$		\xa0
	$3$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(- 2)$ and place the result of $(20)$ under the next term in the dividend $(- 1)$ .

$- 2$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 6$	$20$	\xa0
	$3$	$- 10$

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

$- 2$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 6$	$20$	\xa0
	$3$	$- 10$	$19$

Multiply the newest entry in the result $(19)$ by the divisor $(- 2)$ and place the result of $(- 38)$ under the next term in the dividend $(6)$ .

$- 2$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 6$	$20$	$- 38$
	$3$	$- 10$	$19$

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

$- 2$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 6$	$20$	$- 38$
	$3$	$- 10$	$19$	$- 32$

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

$3 x^{2} + - 10 x + 19 + \frac{- 32}{x + 2}$

Simplify the quotient polynomial.

$3 x^{2} - 10 x + 19 - \frac{32}{x + 2}$

Step 7

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 8

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

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

$3$	$3$	$- 4$	$- 1$	$6$
	\xa0			\xa0

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

$3$	$3$	$- 4$	$- 1$	$6$
	\xa0			\xa0
	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(3)$ and place the result of $(9)$ under the next term in the dividend $(- 4)$ .

$3$	$3$	$- 4$	$- 1$	$6$
	\xa0	$9$		\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.

$3$	$3$	$- 4$	$- 1$	$6$
	\xa0	$9$		\xa0
	$3$	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(3)$ and place the result of $(15)$ under the next term in the dividend $(- 1)$ .

$3$	$3$	$- 4$	$- 1$	$6$
	\xa0	$9$	$15$	\xa0
	$3$	$5$

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

$3$	$3$	$- 4$	$- 1$	$6$
	\xa0	$9$	$15$	\xa0
	$3$	$5$	$14$

Multiply the newest entry in the result $(14)$ by the divisor $(3)$ and place the result of $(42)$ under the next term in the dividend $(6)$ .

$3$	$3$	$- 4$	$- 1$	$6$
	\xa0	$9$	$15$	$42$
	$3$	$5$	$14$

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

$3$	$3$	$- 4$	$- 1$	$6$
	\xa0	$9$	$15$	$42$
	$3$	$5$	$14$	$48$

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

$3 x^{2} + 5 x + 14 + \frac{48}{x - 3}$

Step 9

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 10

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

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

$- 3$	$3$	$- 4$	$- 1$	$6$
	\xa0			\xa0

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

$- 3$	$3$	$- 4$	$- 1$	$6$
	\xa0			\xa0
	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(- 3)$ and place the result of $(- 9)$ under the next term in the dividend $(- 4)$ .

$- 3$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 9$		\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.

$- 3$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 9$		\xa0
	$3$	$- 13$

Multiply the newest entry in the result $(- 13)$ by the divisor $(- 3)$ and place the result of $(39)$ under the next term in the dividend $(- 1)$ .

$- 3$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 9$	$39$	\xa0
	$3$	$- 13$

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

$- 3$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 9$	$39$	\xa0
	$3$	$- 13$	$38$

Multiply the newest entry in the result $(38)$ by the divisor $(- 3)$ and place the result of $(- 114)$ under the next term in the dividend $(6)$ .

$- 3$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 9$	$39$	$- 114$
	$3$	$- 13$	$38$

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

$- 3$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 9$	$39$	$- 114$
	$3$	$- 13$	$38$	$- 108$

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

$3 x^{2} + - 13 x + 38 + \frac{- 108}{x + 3}$

Simplify the quotient polynomial.

$3 x^{2} - 13 x + 38 - \frac{108}{x + 3}$

Step 11

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

Lower Bound: $- 3$

Step 12

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

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

$6$	$3$	$- 4$	$- 1$	$6$
	\xa0			\xa0

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

$6$	$3$	$- 4$	$- 1$	$6$
	\xa0			\xa0
	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(6)$ and place the result of $(18)$ under the next term in the dividend $(- 4)$ .

$6$	$3$	$- 4$	$- 1$	$6$
	\xa0	$18$		\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.

$6$	$3$	$- 4$	$- 1$	$6$
	\xa0	$18$		\xa0
	$3$	$14$

Multiply the newest entry in the result $(14)$ by the divisor $(6)$ and place the result of $(84)$ under the next term in the dividend $(- 1)$ .

$6$	$3$	$- 4$	$- 1$	$6$
	\xa0	$18$	$84$	\xa0
	$3$	$14$

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

$6$	$3$	$- 4$	$- 1$	$6$
	\xa0	$18$	$84$	\xa0
	$3$	$14$	$83$

Multiply the newest entry in the result $(83)$ by the divisor $(6)$ and place the result of $(498)$ under the next term in the dividend $(6)$ .

$6$	$3$	$- 4$	$- 1$	$6$
	\xa0	$18$	$84$	$498$
	$3$	$14$	$83$

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

$6$	$3$	$- 4$	$- 1$	$6$
	\xa0	$18$	$84$	$498$
	$3$	$14$	$83$	$504$

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

$3 x^{2} + 14 x + 83 + \frac{504}{x - 6}$

Step 13

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 14

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

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

$- 6$	$3$	$- 4$	$- 1$	$6$
	\xa0			\xa0

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

$- 6$	$3$	$- 4$	$- 1$	$6$
	\xa0			\xa0
	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(- 6)$ and place the result of $(- 18)$ under the next term in the dividend $(- 4)$ .

$- 6$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 18$		\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.

$- 6$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 18$		\xa0
	$3$	$- 22$

Multiply the newest entry in the result $(- 22)$ by the divisor $(- 6)$ and place the result of $(132)$ under the next term in the dividend $(- 1)$ .

$- 6$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 18$	$132$	\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.

$- 6$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 18$	$132$	\xa0
	$3$	$- 22$	$131$

Multiply the newest entry in the result $(131)$ by the divisor $(- 6)$ and place the result of $(- 786)$ under the next term in the dividend $(6)$ .

$- 6$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 18$	$132$	$- 786$
	$3$	$- 22$	$131$

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

$- 6$	$3$	$- 4$	$- 1$	$6$
	\xa0	$- 18$	$132$	$- 786$
	$3$	$- 22$	$131$	$- 780$

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

$3 x^{2} + - 22 x + 131 + \frac{- 780}{x + 6}$

Simplify the quotient polynomial.

$3 x^{2} - 22 x + 131 - \frac{780}{x + 6}$

Step 15

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 16

Determine the upper and lower bounds.

Upper Bounds: $2, 3, 6$

Lower Bounds: $- 1, - 2, - 3, - 6$

Step 17

Find the Upper and Lower Bounds 3x^3-4x^2-x+6

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