Find the Upper and Lower Bounds G(x)=x^3-x^2-3x+3

$G (x) = x^{3} - x^{2} - 3 x + 3$

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.

$p = \pm 1, \pm 3$

$q = \pm 1$

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

$\pm 1, \pm 3$

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

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

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

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

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

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

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

Multiply the newest entry in the result $(2)$ by the divisor $(3)$ and place the result of $(6)$ under the next term in the dividend $(- 3)$ .

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

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

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

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 $(3)$ .

$3$	$1$	$- 1$	$- 3$	$3$
	\xa0	$3$	$6$	$9$
	$1$	$2$	$3$

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

$3$	$1$	$- 1$	$- 3$	$3$
	\xa0	$3$	$6$	$9$
	$1$	$2$	$3$	$12$

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

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

Simplify the quotient polynomial.

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

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: $3$

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

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

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

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

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

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

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

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

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

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

$- 3$	$1$	$- 1$	$- 3$	$3$
	\xa0	$- 3$	$12$	\xa0
	$1$	$- 4$	$9$

Multiply the newest entry in the result $(9)$ by the divisor $(- 3)$ and place the result of $(- 27)$ under the next term in the dividend $(3)$ .

$- 3$	$1$	$- 1$	$- 3$	$3$
	\xa0	$- 3$	$12$	$- 27$
	$1$	$- 4$	$9$

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

$- 3$	$1$	$- 1$	$- 3$	$3$
	\xa0	$- 3$	$12$	$- 27$
	$1$	$- 4$	$9$	$- 24$

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

$1 x^{2} + - 4 x + 9 + \frac{- 24}{x + 3}$

Simplify the quotient polynomial.

$x^{2} - 4 x + 9 - \frac{24}{x + 3}$

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: $- 3$

Determine the upper and lower bounds.

Upper Bound: $3$

Lower Bound: $- 3$

Find the Upper and Lower Bounds G(x)=x^3-x^2-3x+3

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