Find the Upper and Lower Bounds f(x)=2x^4-7x^3+x^2+7x-3

$f (x) = 2 x^{4} - 7 x^{3} + x^{2} + 7 x - 3$

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.

$p = \pm 1, \pm 3$

$q = \pm 1, \pm 2$

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

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

Apply synthetic division on $\frac{2 x^{4} - 7 x^{3} + x^{2} + 7 x - 3}{x + 1}$ when $x = - 1$ .

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

$- 1$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0				\xa0

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0				\xa0
	$2$

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 $(- 7)$ .

$- 1$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 2$			\xa0
	$2$

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

$- 1$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 2$			\xa0
	$2$	$- 9$

Multiply the newest entry in the result $(- 9)$ by the divisor $(- 1)$ and place the result of $(9)$ under the next term in the dividend $(1)$ .

$- 1$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 2$	$9$		\xa0
	$2$	$- 9$

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

$- 1$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 2$	$9$		\xa0
	$2$	$- 9$	$10$

Multiply the newest entry in the result $(10)$ by the divisor $(- 1)$ and place the result of $(- 10)$ under the next term in the dividend $(7)$ .

$- 1$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 2$	$9$	$- 10$	\xa0
	$2$	$- 9$	$10$

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

$- 1$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 2$	$9$	$- 10$	\xa0
	$2$	$- 9$	$10$	$- 3$

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 $(- 3)$ .

$- 1$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 2$	$9$	$- 10$	$3$
	$2$	$- 9$	$10$	$- 3$

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

$- 1$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 2$	$9$	$- 10$	$3$
	$2$	$- 9$	$10$	$- 3$	$0$

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

$2 x^{3} + - 9 x^{2} + (10) x - 3$

Simplify the quotient polynomial.

$2 x^{3} - 9 x^{2} + 10 x - 3$

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: $- 1$

Apply synthetic division on $\frac{2 x^{4} - 7 x^{3} + x^{2} + 7 x - 3}{x + 3}$ when $x = - 3$ .

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

$- 3$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0				\xa0

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- 3$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0				\xa0
	$2$

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 $(- 7)$ .

$- 3$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 6$			\xa0
	$2$

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

$- 3$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 6$			\xa0
	$2$	$- 13$

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 $(1)$ .

$- 3$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 6$	$39$		\xa0
	$2$	$- 13$

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

$- 3$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 6$	$39$		\xa0
	$2$	$- 13$	$40$

Multiply the newest entry in the result $(40)$ by the divisor $(- 3)$ and place the result of $(- 120)$ under the next term in the dividend $(7)$ .

$- 3$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 6$	$39$	$- 120$	\xa0
	$2$	$- 13$	$40$

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

$- 3$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 6$	$39$	$- 120$	\xa0
	$2$	$- 13$	$40$	$- 113$

Multiply the newest entry in the result $(- 113)$ by the divisor $(- 3)$ and place the result of $(339)$ under the next term in the dividend $(- 3)$ .

$- 3$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 6$	$39$	$- 120$	$339$
	$2$	$- 13$	$40$	$- 113$

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

$- 3$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 6$	$39$	$- 120$	$339$
	$2$	$- 13$	$40$	$- 113$	$336$

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

$2 x^{3} + - 13 x^{2} + (40) x - 113 + \frac{336}{x + 3}$

Simplify the quotient polynomial.

$2 x^{3} - 13 x^{2} + 40 x - 113 + \frac{336}{x + 3}$

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: $- 3$

Apply synthetic division on $\frac{2 x^{4} - 7 x^{3} + x^{2} + 7 x - 3}{x + \frac{3}{2}}$ when $x = - \frac{3}{2}$ .

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

$- \frac{3}{2}$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0				\xa0

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- \frac{3}{2}$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0				\xa0
	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(- \frac{3}{2})$ and place the result of $(- 3)$ under the next term in the dividend $(- 7)$ .

$- \frac{3}{2}$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 3$			\xa0
	$2$

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

$- \frac{3}{2}$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 3$			\xa0
	$2$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(- \frac{3}{2})$ and place the result of $(15)$ under the next term in the dividend $(1)$ .

$- \frac{3}{2}$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 3$	$15$		\xa0
	$2$	$- 10$

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

$- \frac{3}{2}$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 3$	$15$		\xa0
	$2$	$- 10$	$16$

Multiply the newest entry in the result $(16)$ by the divisor $(- \frac{3}{2})$ and place the result of $(- 24)$ under the next term in the dividend $(7)$ .

$- \frac{3}{2}$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 3$	$15$	$- 24$	\xa0
	$2$	$- 10$	$16$

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

$- \frac{3}{2}$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 3$	$15$	$- 24$	\xa0
	$2$	$- 10$	$16$	$- 17$

Multiply the newest entry in the result $(- 17)$ by the divisor $(- \frac{3}{2})$ and place the result of $(\frac{51}{2})$ under the next term in the dividend $(- 3)$ .

$- \frac{3}{2}$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 3$	$15$	$- 24$	$\frac{51}{2}$
	$2$	$- 10$	$16$	$- 17$

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

$- \frac{3}{2}$	$2$	$- 7$	$1$	$7$	$- 3$
	\xa0	$- 3$	$15$	$- 24$	$\frac{51}{2}$
	$2$	$- 10$	$16$	$- 17$	$\frac{45}{2}$

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

$2 x^{3} + - 10 x^{2} + (16) x - 17 + \frac{\frac{45}{2}}{x + \frac{3}{2}}$

Simplify the quotient polynomial.

$2 x^{3} - 10 x^{2} + 16 x - 17 + \frac{45}{2 x + 3}$

Since $- \frac{3}{2} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{3}{2}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{3}{2}$

Determine the upper and lower bounds.

No Upper Bounds

Lower Bounds: $- 1, - 3, - \frac{3}{2}$

Find the Upper and Lower Bounds f(x)=2x^4-7x^3+x^2+7x-3

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