Find the Upper and Lower Bounds 3x^4-5x^3-5x^2+5x+2

$3 x^{4} - 5 x^{3} - 5 x^{2} + 5 x + 2$

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 3$

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

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

Step 2

Apply synthetic division on $\frac{3 x^{4} - 5 x^{3} - 5 x^{2} + 5 x + 2}{x + 2}$ when $x = - 2$ .

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

$- 2$	$3$	$- 5$	$- 5$	$5$	$2$
	\xa0				\xa0

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

$- 2$	$3$	$- 5$	$- 5$	$5$	$2$
	\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 $(- 5)$ .

$- 2$	$3$	$- 5$	$- 5$	$5$	$2$
	\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.

$- 2$	$3$	$- 5$	$- 5$	$5$	$2$
	\xa0	$- 6$			\xa0
	$3$	$- 11$

Multiply the newest entry in the result $(- 11)$ by the divisor $(- 2)$ and place the result of $(22)$ under the next term in the dividend $(- 5)$ .

$- 2$	$3$	$- 5$	$- 5$	$5$	$2$
	\xa0	$- 6$	$22$		\xa0
	$3$	$- 11$

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

$- 2$	$3$	$- 5$	$- 5$	$5$	$2$
	\xa0	$- 6$	$22$		\xa0
	$3$	$- 11$	$17$

Multiply the newest entry in the result $(17)$ by the divisor $(- 2)$ and place the result of $(- 34)$ under the next term in the dividend $(5)$ .

$- 2$	$3$	$- 5$	$- 5$	$5$	$2$
	\xa0	$- 6$	$22$	$- 34$	\xa0
	$3$	$- 11$	$17$

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

$- 2$	$3$	$- 5$	$- 5$	$5$	$2$
	\xa0	$- 6$	$22$	$- 34$	\xa0
	$3$	$- 11$	$17$	$- 29$

Multiply the newest entry in the result $(- 29)$ by the divisor $(- 2)$ and place the result of $(58)$ under the next term in the dividend $(2)$ .

$- 2$	$3$	$- 5$	$- 5$	$5$	$2$
	\xa0	$- 6$	$22$	$- 34$	$58$
	$3$	$- 11$	$17$	$- 29$

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

$- 2$	$3$	$- 5$	$- 5$	$5$	$2$
	\xa0	$- 6$	$22$	$- 34$	$58$
	$3$	$- 11$	$17$	$- 29$	$60$

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

$3 x^{3} + - 11 x^{2} + (17) x - 29 + \frac{60}{x + 2}$

Simplify the quotient polynomial.

$3 x^{3} - 11 x^{2} + 17 x - 29 + \frac{60}{x + 2}$

Step 3

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 4

Determine the upper and lower bounds.

No Upper Bounds

Lower Bound: $- 2$

Step 5

Find the Upper and Lower Bounds 3x^4-5x^3-5x^2+5x+2

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