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

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

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.

$p = \pm 1, \pm 3$

$q = \pm 1$

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

$\pm 1, \pm 3$

Step 2

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

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

$3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0				\xa0

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

$3$	$1$	$0$	$- 3$	$0$	$- 3$
	\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 $(0)$ .

$3$	$1$	$0$	$- 3$	$0$	$- 3$
	\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.

$3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$3$			\xa0
	$1$	$3$

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 $(- 3)$ .

$3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$3$	$9$		\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.

$3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$3$	$9$		\xa0
	$1$	$3$	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(3)$ and place the result of $(18)$ under the next term in the dividend $(0)$ .

$3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$3$	$9$	$18$	\xa0
	$1$	$3$	$6$

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

$3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$3$	$9$	$18$	\xa0
	$1$	$3$	$6$	$18$

Multiply the newest entry in the result $(18)$ by the divisor $(3)$ and place the result of $(54)$ under the next term in the dividend $(- 3)$ .

$3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$3$	$9$	$18$	$54$
	$1$	$3$	$6$	$18$

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

$3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$3$	$9$	$18$	$54$
	$1$	$3$	$6$	$18$	$51$

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

$1 x^{3} + 3 x^{2} + (6) x + 18 + \frac{51}{x - 3}$

Simplify the quotient polynomial.

$x^{3} + 3 x^{2} + 6 x + 18 + \frac{51}{x - 3}$

Step 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: $3$

Step 4

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

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

$- 3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0				\xa0

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

$- 3$	$1$	$0$	$- 3$	$0$	$- 3$
	\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 $(0)$ .

$- 3$	$1$	$0$	$- 3$	$0$	$- 3$
	\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.

$- 3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$- 3$			\xa0
	$1$	$- 3$

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 $(- 3)$ .

$- 3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$- 3$	$9$		\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.

$- 3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$- 3$	$9$		\xa0
	$1$	$- 3$	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(- 3)$ and place the result of $(- 18)$ under the next term in the dividend $(0)$ .

$- 3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$- 3$	$9$	$- 18$	\xa0
	$1$	$- 3$	$6$

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

$- 3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$- 3$	$9$	$- 18$	\xa0
	$1$	$- 3$	$6$	$- 18$

Multiply the newest entry in the result $(- 18)$ by the divisor $(- 3)$ and place the result of $(54)$ under the next term in the dividend $(- 3)$ .

$- 3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$- 3$	$9$	$- 18$	$54$
	$1$	$- 3$	$6$	$- 18$

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

$- 3$	$1$	$0$	$- 3$	$0$	$- 3$
	\xa0	$- 3$	$9$	$- 18$	$54$
	$1$	$- 3$	$6$	$- 18$	$51$

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

$1 x^{3} + - 3 x^{2} + (6) x - 18 + \frac{51}{x + 3}$

Simplify the quotient polynomial.

$x^{3} - 3 x^{2} + 6 x - 18 + \frac{51}{x + 3}$

Step 5

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 6

Determine the upper and lower bounds.

Upper Bound: $3$

Lower Bound: $- 3$

Step 7

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

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