Find the Upper and Lower Bounds f(x)=x^3-2x^2-7x+14

$f (x) = x^{3} - 2 x^{2} - 7 x + 14$

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.

$p = \pm 1, \pm 2, \pm 7, \pm 14$

$q = \pm 1$

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

$\pm 1, \pm 2, \pm 7, \pm 14$

Apply synthetic division on $\frac{x^{3} - 2 x^{2} - 7 x + 14}{x - 7}$ when $x = 7$ .

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

$7$	$1$	$- 2$	$- 7$	$14$
	\xa0			\xa0

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

$7$	$1$	$- 2$	$- 7$	$14$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(7)$ and place the result of $(7)$ under the next term in the dividend $(- 2)$ .

$7$	$1$	$- 2$	$- 7$	$14$
	\xa0	$7$		\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.

$7$	$1$	$- 2$	$- 7$	$14$
	\xa0	$7$		\xa0
	$1$	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(7)$ and place the result of $(35)$ under the next term in the dividend $(- 7)$ .

$7$	$1$	$- 2$	$- 7$	$14$
	\xa0	$7$	$35$	\xa0
	$1$	$5$

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

$7$	$1$	$- 2$	$- 7$	$14$
	\xa0	$7$	$35$	\xa0
	$1$	$5$	$28$

Multiply the newest entry in the result $(28)$ by the divisor $(7)$ and place the result of $(196)$ under the next term in the dividend $(14)$ .

$7$	$1$	$- 2$	$- 7$	$14$
	\xa0	$7$	$35$	$196$
	$1$	$5$	$28$

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

$7$	$1$	$- 2$	$- 7$	$14$
	\xa0	$7$	$35$	$196$
	$1$	$5$	$28$	$210$

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

$1 x^{2} + 5 x + 28 + \frac{210}{x - 7}$

Simplify the quotient polynomial.

$x^{2} + 5 x + 28 + \frac{210}{x - 7}$

Since $7 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $7$ is an upper bound for the real roots of the function.

Upper Bound: $7$

Apply synthetic division on $\frac{x^{3} - 2 x^{2} - 7 x + 14}{x + 7}$ when $x = - 7$ .

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

$- 7$	$1$	$- 2$	$- 7$	$14$
	\xa0			\xa0

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

$- 7$	$1$	$- 2$	$- 7$	$14$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 7)$ and place the result of $(- 7)$ under the next term in the dividend $(- 2)$ .

$- 7$	$1$	$- 2$	$- 7$	$14$
	\xa0	$- 7$		\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.

$- 7$	$1$	$- 2$	$- 7$	$14$
	\xa0	$- 7$		\xa0
	$1$	$- 9$

Multiply the newest entry in the result $(- 9)$ by the divisor $(- 7)$ and place the result of $(63)$ under the next term in the dividend $(- 7)$ .

$- 7$	$1$	$- 2$	$- 7$	$14$
	\xa0	$- 7$	$63$	\xa0
	$1$	$- 9$

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

$- 7$	$1$	$- 2$	$- 7$	$14$
	\xa0	$- 7$	$63$	\xa0
	$1$	$- 9$	$56$

Multiply the newest entry in the result $(56)$ by the divisor $(- 7)$ and place the result of $(- 392)$ under the next term in the dividend $(14)$ .

$- 7$	$1$	$- 2$	$- 7$	$14$
	\xa0	$- 7$	$63$	$- 392$
	$1$	$- 9$	$56$

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

$- 7$	$1$	$- 2$	$- 7$	$14$
	\xa0	$- 7$	$63$	$- 392$
	$1$	$- 9$	$56$	$- 378$

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

$1 x^{2} + - 9 x + 56 + \frac{- 378}{x + 7}$

Simplify the quotient polynomial.

$x^{2} - 9 x + 56 - \frac{378}{x + 7}$

Since $- 7 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 7$ is a lower bound for the real roots of the function.

Lower Bound: $- 7$

Apply synthetic division on $\frac{x^{3} - 2 x^{2} - 7 x + 14}{x - 14}$ when $x = 14$ .

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

$14$	$1$	$- 2$	$- 7$	$14$
	\xa0			\xa0

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

$14$	$1$	$- 2$	$- 7$	$14$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(14)$ and place the result of $(14)$ under the next term in the dividend $(- 2)$ .

$14$	$1$	$- 2$	$- 7$	$14$
	\xa0	$14$		\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.

$14$	$1$	$- 2$	$- 7$	$14$
	\xa0	$14$		\xa0
	$1$	$12$

Multiply the newest entry in the result $(12)$ by the divisor $(14)$ and place the result of $(168)$ under the next term in the dividend $(- 7)$ .

$14$	$1$	$- 2$	$- 7$	$14$
	\xa0	$14$	$168$	\xa0
	$1$	$12$

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

$14$	$1$	$- 2$	$- 7$	$14$
	\xa0	$14$	$168$	\xa0
	$1$	$12$	$161$

Multiply the newest entry in the result $(161)$ by the divisor $(14)$ and place the result of $(2254)$ under the next term in the dividend $(14)$ .

$14$	$1$	$- 2$	$- 7$	$14$
	\xa0	$14$	$168$	$2254$
	$1$	$12$	$161$

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

$14$	$1$	$- 2$	$- 7$	$14$
	\xa0	$14$	$168$	$2254$
	$1$	$12$	$161$	$2268$

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

$1 x^{2} + 12 x + 161 + \frac{2268}{x - 14}$

Simplify the quotient polynomial.

$x^{2} + 12 x + 161 + \frac{2268}{x - 14}$

Since $14 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $14$ is an upper bound for the real roots of the function.

Upper Bound: $14$

Apply synthetic division on $\frac{x^{3} - 2 x^{2} - 7 x + 14}{x + 14}$ when $x = - 14$ .

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

$- 14$	$1$	$- 2$	$- 7$	$14$
	\xa0			\xa0

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

$- 14$	$1$	$- 2$	$- 7$	$14$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 14)$ and place the result of $(- 14)$ under the next term in the dividend $(- 2)$ .

$- 14$	$1$	$- 2$	$- 7$	$14$
	\xa0	$- 14$		\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.

$- 14$	$1$	$- 2$	$- 7$	$14$
	\xa0	$- 14$		\xa0
	$1$	$- 16$

Multiply the newest entry in the result $(- 16)$ by the divisor $(- 14)$ and place the result of $(224)$ under the next term in the dividend $(- 7)$ .

$- 14$	$1$	$- 2$	$- 7$	$14$
	\xa0	$- 14$	$224$	\xa0
	$1$	$- 16$

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

$- 14$	$1$	$- 2$	$- 7$	$14$
	\xa0	$- 14$	$224$	\xa0
	$1$	$- 16$	$217$

Multiply the newest entry in the result $(217)$ by the divisor $(- 14)$ and place the result of $(- 3038)$ under the next term in the dividend $(14)$ .

$- 14$	$1$	$- 2$	$- 7$	$14$
	\xa0	$- 14$	$224$	$- 3038$
	$1$	$- 16$	$217$

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

$- 14$	$1$	$- 2$	$- 7$	$14$
	\xa0	$- 14$	$224$	$- 3038$
	$1$	$- 16$	$217$	$- 3024$

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

$1 x^{2} + - 16 x + 217 + \frac{- 3024}{x + 14}$

Simplify the quotient polynomial.

$x^{2} - 16 x + 217 - \frac{3024}{x + 14}$

Since $- 14 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 14$ is a lower bound for the real roots of the function.

Lower Bound: $- 14$

Determine the upper and lower bounds.

Upper Bounds: $7, 14$

Lower Bounds: $- 7, - 14$

Find the Upper and Lower Bounds f(x)=x^3-2x^2-7x+14

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