Find the Upper and Lower Bounds 3x^3-x^2+12x-3

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

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.

$p = \pm 1, \pm 3$

$q = \pm 1, \pm 3$

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

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

Step 2

Apply synthetic division on $\frac{3 x^{3} - x^{2} + 12 x - 3}{x - 1}$ when $x = 1$ .

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

$1$	$3$	$- 1$	$12$	$- 3$
	\xa0			\xa0

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

$1$	$3$	$- 1$	$12$	$- 3$
	\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 $(- 1)$ .

$1$	$3$	$- 1$	$12$	$- 3$
	\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.

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

Multiply the newest entry in the result $(2)$ by the divisor $(1)$ and place the result of $(2)$ under the next term in the dividend $(12)$ .

$1$	$3$	$- 1$	$12$	$- 3$
	\xa0	$3$	$2$	\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.

$1$	$3$	$- 1$	$12$	$- 3$
	\xa0	$3$	$2$	\xa0
	$3$	$2$	$14$

Multiply the newest entry in the result $(14)$ by the divisor $(1)$ and place the result of $(14)$ under the next term in the dividend $(- 3)$ .

$1$	$3$	$- 1$	$12$	$- 3$
	\xa0	$3$	$2$	$14$
	$3$	$2$	$14$

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

$1$	$3$	$- 1$	$12$	$- 3$
	\xa0	$3$	$2$	$14$
	$3$	$2$	$14$	$11$

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

$3 x^{2} + 2 x + 14 + \frac{11}{x - 1}$

Step 3

Since $1 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $1$ is an upper bound for the real roots of the function.

Upper Bound: $1$

Step 4

Apply synthetic division on $\frac{3 x^{3} - x^{2} + 12 x - 3}{x + 1}$ when $x = - 1$ .

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

$- 1$	$3$	$- 1$	$12$	$- 3$
	\xa0			\xa0

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

$- 1$	$3$	$- 1$	$12$	$- 3$
	\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 $(- 1)$ .

$- 1$	$3$	$- 1$	$12$	$- 3$
	\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.

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

Multiply the newest entry in the result $(- 4)$ by the divisor $(- 1)$ and place the result of $(4)$ under the next term in the dividend $(12)$ .

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

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

$- 1$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 3$	$4$	\xa0
	$3$	$- 4$	$16$

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

$- 1$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 3$	$4$	$- 16$
	$3$	$- 4$	$16$

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

$- 1$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 3$	$4$	$- 16$
	$3$	$- 4$	$16$	$- 19$

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

$3 x^{2} + - 4 x + 16 + \frac{- 19}{x + 1}$

Simplify the quotient polynomial.

$3 x^{2} - 4 x + 16 - \frac{19}{x + 1}$

Step 5

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 6

Apply synthetic division on $\frac{3 x^{3} - x^{2} + 12 x - 3}{x - \frac{1}{3}}$ when $x = \frac{1}{3}$ .

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

$\frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0			\xa0

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

$\frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0			\xa0
	$3$

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

$\frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0	$1$		\xa0
	$3$

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

$\frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0	$1$		\xa0
	$3$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(\frac{1}{3})$ and place the result of $(0)$ under the next term in the dividend $(12)$ .

$\frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0	$1$	$0$	\xa0
	$3$	$0$

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

$\frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0	$1$	$0$	\xa0
	$3$	$0$	$12$

Multiply the newest entry in the result $(12)$ by the divisor $(\frac{1}{3})$ and place the result of $(4)$ under the next term in the dividend $(- 3)$ .

$\frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0	$1$	$0$	$4$
	$3$	$0$	$12$

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

$\frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0	$1$	$0$	$4$
	$3$	$0$	$12$	$1$

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

$3 x^{2} + 0 x + 12 + \frac{1}{x - \frac{1}{3}}$

Simplify the quotient polynomial.

$3 x^{2} + 12 + \frac{3}{3 x - 1}$

Step 7

Since $\frac{1}{3} > 0$ and all of the signs in the bottom row of the synthetic division are positive, $\frac{1}{3}$ is an upper bound for the real roots of the function.

Upper Bound: $\frac{1}{3}$

Step 8

Apply synthetic division on $\frac{3 x^{3} - x^{2} + 12 x - 3}{x + \frac{1}{3}}$ when $x = - \frac{1}{3}$ .

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

$- \frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0			\xa0

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

$- \frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0			\xa0
	$3$

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

$- \frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 1$		\xa0
	$3$

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

$- \frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 1$		\xa0
	$3$	$- 2$

Multiply the newest entry in the result $(- 2)$ by the divisor $(- \frac{1}{3})$ and place the result of $(\frac{2}{3})$ under the next term in the dividend $(12)$ .

$- \frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 1$	$\frac{2}{3}$	\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.

$- \frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 1$	$\frac{2}{3}$	\xa0
	$3$	$- 2$	$\frac{38}{3}$

Multiply the newest entry in the result $(\frac{38}{3})$ by the divisor $(- \frac{1}{3})$ and place the result of $(- \frac{38}{9})$ under the next term in the dividend $(- 3)$ .

$- \frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 1$	$\frac{2}{3}$	$- \frac{38}{9}$
	$3$	$- 2$	$\frac{38}{3}$

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

$- \frac{1}{3}$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 1$	$\frac{2}{3}$	$- \frac{38}{9}$
	$3$	$- 2$	$\frac{38}{3}$	$- \frac{65}{9}$

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

$3 x^{2} + - 2 x + \frac{38}{3} + \frac{- \frac{65}{9}}{x + \frac{1}{3}}$

Simplify the quotient polynomial.

$3 x^{2} - 2 x + \frac{38}{3} - \frac{65}{3 (3 x + 1)}$

Step 9

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

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

Step 10

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

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

$3$	$3$	$- 1$	$12$	$- 3$
	\xa0			\xa0

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

$3$	$3$	$- 1$	$12$	$- 3$
	\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 $(- 1)$ .

$3$	$3$	$- 1$	$12$	$- 3$
	\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.

$3$	$3$	$- 1$	$12$	$- 3$
	\xa0	$9$		\xa0
	$3$	$8$

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 $(12)$ .

$3$	$3$	$- 1$	$12$	$- 3$
	\xa0	$9$	$24$	\xa0
	$3$	$8$

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

$3$	$3$	$- 1$	$12$	$- 3$
	\xa0	$9$	$24$	\xa0
	$3$	$8$	$36$

Multiply the newest entry in the result $(36)$ by the divisor $(3)$ and place the result of $(108)$ under the next term in the dividend $(- 3)$ .

$3$	$3$	$- 1$	$12$	$- 3$
	\xa0	$9$	$24$	$108$
	$3$	$8$	$36$

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

$3$	$3$	$- 1$	$12$	$- 3$
	\xa0	$9$	$24$	$108$
	$3$	$8$	$36$	$105$

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

$3 x^{2} + 8 x + 36 + \frac{105}{x - 3}$

Step 11

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 12

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

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

$- 3$	$3$	$- 1$	$12$	$- 3$
	\xa0			\xa0

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

$- 3$	$3$	$- 1$	$12$	$- 3$
	\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 $(- 1)$ .

$- 3$	$3$	$- 1$	$12$	$- 3$
	\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.

$- 3$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 9$		\xa0
	$3$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(- 3)$ and place the result of $(30)$ under the next term in the dividend $(12)$ .

$- 3$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 9$	$30$	\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.

$- 3$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 9$	$30$	\xa0
	$3$	$- 10$	$42$

Multiply the newest entry in the result $(42)$ by the divisor $(- 3)$ and place the result of $(- 126)$ under the next term in the dividend $(- 3)$ .

$- 3$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 9$	$30$	$- 126$
	$3$	$- 10$	$42$

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

$- 3$	$3$	$- 1$	$12$	$- 3$
	\xa0	$- 9$	$30$	$- 126$
	$3$	$- 10$	$42$	$- 129$

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

$3 x^{2} + - 10 x + 42 + \frac{- 129}{x + 3}$

Simplify the quotient polynomial.

$3 x^{2} - 10 x + 42 - \frac{129}{x + 3}$

Step 13

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 14

Determine the upper and lower bounds.

Upper Bounds: $1, \frac{1}{3}, 3$

Lower Bounds: $- 1, - \frac{1}{3}, - 3$

Step 15

Find the Upper and Lower Bounds 3x^3-x^2+12x-3

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