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

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

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.

$p = \pm 1, \pm 7$

$q = \pm 1$

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

$\pm 1, \pm 7$

Step 2

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

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

$- 1$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0				\xa0

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

$- 1$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0				\xa0
	$1$

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

$- 1$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 1$			\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.

$- 1$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 1$			\xa0
	$1$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(- 1)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$- 1$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 1$	$0$		\xa0
	$1$	$0$

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

$- 1$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 1$	$0$		\xa0
	$1$	$0$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(- 1)$ and place the result of $(0)$ under the next term in the dividend $(- 7)$ .

$- 1$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 1$	$0$	$0$	\xa0
	$1$	$0$	$0$

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

$- 1$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 1$	$0$	$0$	\xa0
	$1$	$0$	$0$	$- 7$

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

$- 1$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 1$	$0$	$0$	$7$
	$1$	$0$	$0$	$- 7$

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

$- 1$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 1$	$0$	$0$	$7$
	$1$	$0$	$0$	$- 7$	$0$

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

$1 x^{3} + 0 x^{2} + (0) x - 7$

Simplify the quotient polynomial.

$x^{3} - 7$

Step 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: $- 1$

Step 4

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

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

$7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0				\xa0

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

$7$	$1$	$1$	$0$	$- 7$	$- 7$
	\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 $(1)$ .

$7$	$1$	$1$	$0$	$- 7$	$- 7$
	\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.

$7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$7$			\xa0
	$1$	$8$

Multiply the newest entry in the result $(8)$ by the divisor $(7)$ and place the result of $(56)$ under the next term in the dividend $(0)$ .

$7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$7$	$56$		\xa0
	$1$	$8$

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

$7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$7$	$56$		\xa0
	$1$	$8$	$56$

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 $(- 7)$ .

$7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$7$	$56$	$392$	\xa0
	$1$	$8$	$56$

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

$7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$7$	$56$	$392$	\xa0
	$1$	$8$	$56$	$385$

Multiply the newest entry in the result $(385)$ by the divisor $(7)$ and place the result of $(2695)$ under the next term in the dividend $(- 7)$ .

$7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$7$	$56$	$392$	$2695$
	$1$	$8$	$56$	$385$

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

$7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$7$	$56$	$392$	$2695$
	$1$	$8$	$56$	$385$	$2688$

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

$1 x^{3} + 8 x^{2} + (56) x + 385 + \frac{2688}{x - 7}$

Simplify the quotient polynomial.

$x^{3} + 8 x^{2} + 56 x + 385 + \frac{2688}{x - 7}$

Step 5

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: $7$

Step 6

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

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

$- 7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0				\xa0

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

$- 7$	$1$	$1$	$0$	$- 7$	$- 7$
	\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 $(1)$ .

$- 7$	$1$	$1$	$0$	$- 7$	$- 7$
	\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.

$- 7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 7$			\xa0
	$1$	$- 6$

Multiply the newest entry in the result $(- 6)$ by the divisor $(- 7)$ and place the result of $(42)$ under the next term in the dividend $(0)$ .

$- 7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 7$	$42$		\xa0
	$1$	$- 6$

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

$- 7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 7$	$42$		\xa0
	$1$	$- 6$	$42$

Multiply the newest entry in the result $(42)$ by the divisor $(- 7)$ and place the result of $(- 294)$ under the next term in the dividend $(- 7)$ .

$- 7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 7$	$42$	$- 294$	\xa0
	$1$	$- 6$	$42$

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

$- 7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 7$	$42$	$- 294$	\xa0
	$1$	$- 6$	$42$	$- 301$

Multiply the newest entry in the result $(- 301)$ by the divisor $(- 7)$ and place the result of $(2107)$ under the next term in the dividend $(- 7)$ .

$- 7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 7$	$42$	$- 294$	$2107$
	$1$	$- 6$	$42$	$- 301$

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

$- 7$	$1$	$1$	$0$	$- 7$	$- 7$
	\xa0	$- 7$	$42$	$- 294$	$2107$
	$1$	$- 6$	$42$	$- 301$	$2100$

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

$1 x^{3} + - 6 x^{2} + (42) x - 301 + \frac{2100}{x + 7}$

Simplify the quotient polynomial.

$x^{3} - 6 x^{2} + 42 x - 301 + \frac{2100}{x + 7}$

Step 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: $- 7$

Step 8

Determine the upper and lower bounds.

Upper Bound: $7$

Lower Bounds: $- 1, - 7$

Step 9

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

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