Find the Upper and Lower Bounds 7x^4+7x^3-x^2-28x-28

$7 x^{4} + 7 x^{3} - x^{2} - 28 x - 28$

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.

$p = \pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28$

$q = \pm 1, \pm 7$

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

$\pm 1, \pm \frac{1}{7}, \pm 2, \pm \frac{2}{7}, \pm 4, \pm \frac{4}{7}, \pm 7, \pm 14, \pm 28$

Step 2

Apply synthetic division on $\frac{7 x^{4} + 7 x^{3} - x^{2} - 28 x - 28}{x - 2}$ when $x = 2$ .

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

$2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0

The first number in the dividend $(7)$ is put into the first position of the result area (below the horizontal line).

$2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0
	$7$

Multiply the newest entry in the result $(7)$ by the divisor $(2)$ and place the result of $(14)$ under the next term in the dividend $(7)$ .

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

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

$2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$14$			\xa0
	$7$	$21$

Multiply the newest entry in the result $(21)$ by the divisor $(2)$ and place the result of $(42)$ under the next term in the dividend $(- 1)$ .

$2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$14$	$42$		\xa0
	$7$	$21$

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

$2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$14$	$42$		\xa0
	$7$	$21$	$41$

Multiply the newest entry in the result $(41)$ by the divisor $(2)$ and place the result of $(82)$ under the next term in the dividend $(- 28)$ .

$2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$14$	$42$	$82$	\xa0
	$7$	$21$	$41$

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

$2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$14$	$42$	$82$	\xa0
	$7$	$21$	$41$	$54$

Multiply the newest entry in the result $(54)$ by the divisor $(2)$ and place the result of $(108)$ under the next term in the dividend $(- 28)$ .

$2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$14$	$42$	$82$	$108$
	$7$	$21$	$41$	$54$

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

$2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$14$	$42$	$82$	$108$
	$7$	$21$	$41$	$54$	$80$

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

$7 x^{3} + 21 x^{2} + (41) x + 54 + \frac{80}{x - 2}$

Simplify the quotient polynomial.

$7 x^{3} + 21 x^{2} + 41 x + 54 + \frac{80}{x - 2}$

Step 3

Since $2 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $2$ is an upper bound for the real roots of the function.

Upper Bound: $2$

Step 4

Apply synthetic division on $\frac{7 x^{4} + 7 x^{3} - x^{2} - 28 x - 28}{x + 2}$ when $x = - 2$ .

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

$- 2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0

The first number in the dividend $(7)$ is put into the first position of the result area (below the horizontal line).

$- 2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0
	$7$

Multiply the newest entry in the result $(7)$ by the divisor $(- 2)$ and place the result of $(- 14)$ under the next term in the dividend $(7)$ .

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

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

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

Multiply the newest entry in the result $(- 7)$ by the divisor $(- 2)$ and place the result of $(14)$ under the next term in the dividend $(- 1)$ .

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

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

$- 2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 14$	$14$		\xa0
	$7$	$- 7$	$13$

Multiply the newest entry in the result $(13)$ by the divisor $(- 2)$ and place the result of $(- 26)$ under the next term in the dividend $(- 28)$ .

$- 2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 14$	$14$	$- 26$	\xa0
	$7$	$- 7$	$13$

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

$- 2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 14$	$14$	$- 26$	\xa0
	$7$	$- 7$	$13$	$- 54$

Multiply the newest entry in the result $(- 54)$ by the divisor $(- 2)$ and place the result of $(108)$ under the next term in the dividend $(- 28)$ .

$- 2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 14$	$14$	$- 26$	$108$
	$7$	$- 7$	$13$	$- 54$

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

$- 2$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 14$	$14$	$- 26$	$108$
	$7$	$- 7$	$13$	$- 54$	$80$

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

$7 x^{3} + - 7 x^{2} + (13) x - 54 + \frac{80}{x + 2}$

Simplify the quotient polynomial.

$7 x^{3} - 7 x^{2} + 13 x - 54 + \frac{80}{x + 2}$

Step 5

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 6

Apply synthetic division on $\frac{7 x^{4} + 7 x^{3} - x^{2} - 28 x - 28}{x - 4}$ when $x = 4$ .

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

$4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0

The first number in the dividend $(7)$ is put into the first position of the result area (below the horizontal line).

$4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0
	$7$

Multiply the newest entry in the result $(7)$ by the divisor $(4)$ and place the result of $(28)$ under the next term in the dividend $(7)$ .

$4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$28$			\xa0
	$7$

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

$4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$28$			\xa0
	$7$	$35$

Multiply the newest entry in the result $(35)$ by the divisor $(4)$ and place the result of $(140)$ under the next term in the dividend $(- 1)$ .

$4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$28$	$140$		\xa0
	$7$	$35$

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

$4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$28$	$140$		\xa0
	$7$	$35$	$139$

Multiply the newest entry in the result $(139)$ by the divisor $(4)$ and place the result of $(556)$ under the next term in the dividend $(- 28)$ .

$4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$28$	$140$	$556$	\xa0
	$7$	$35$	$139$

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

$4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$28$	$140$	$556$	\xa0
	$7$	$35$	$139$	$528$

Multiply the newest entry in the result $(528)$ by the divisor $(4)$ and place the result of $(2112)$ under the next term in the dividend $(- 28)$ .

$4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$28$	$140$	$556$	$2112$
	$7$	$35$	$139$	$528$

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

$4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$28$	$140$	$556$	$2112$
	$7$	$35$	$139$	$528$	$2084$

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

$7 x^{3} + 35 x^{2} + (139) x + 528 + \frac{2084}{x - 4}$

Simplify the quotient polynomial.

$7 x^{3} + 35 x^{2} + 139 x + 528 + \frac{2084}{x - 4}$

Step 7

Since $4 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $4$ is an upper bound for the real roots of the function.

Upper Bound: $4$

Step 8

Apply synthetic division on $\frac{7 x^{4} + 7 x^{3} - x^{2} - 28 x - 28}{x + 4}$ when $x = - 4$ .

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

$- 4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0

The first number in the dividend $(7)$ is put into the first position of the result area (below the horizontal line).

$- 4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0
	$7$

Multiply the newest entry in the result $(7)$ by the divisor $(- 4)$ and place the result of $(- 28)$ under the next term in the dividend $(7)$ .

$- 4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 28$			\xa0
	$7$

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

$- 4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 28$			\xa0
	$7$	$- 21$

Multiply the newest entry in the result $(- 21)$ by the divisor $(- 4)$ and place the result of $(84)$ under the next term in the dividend $(- 1)$ .

$- 4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 28$	$84$		\xa0
	$7$	$- 21$

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

$- 4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 28$	$84$		\xa0
	$7$	$- 21$	$83$

Multiply the newest entry in the result $(83)$ by the divisor $(- 4)$ and place the result of $(- 332)$ under the next term in the dividend $(- 28)$ .

$- 4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 28$	$84$	$- 332$	\xa0
	$7$	$- 21$	$83$

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

$- 4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 28$	$84$	$- 332$	\xa0
	$7$	$- 21$	$83$	$- 360$

Multiply the newest entry in the result $(- 360)$ by the divisor $(- 4)$ and place the result of $(1440)$ under the next term in the dividend $(- 28)$ .

$- 4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 28$	$84$	$- 332$	$1440$
	$7$	$- 21$	$83$	$- 360$

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

$- 4$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 28$	$84$	$- 332$	$1440$
	$7$	$- 21$	$83$	$- 360$	$1412$

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

$7 x^{3} + - 21 x^{2} + (83) x - 360 + \frac{1412}{x + 4}$

Simplify the quotient polynomial.

$7 x^{3} - 21 x^{2} + 83 x - 360 + \frac{1412}{x + 4}$

Step 9

Since $- 4 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 4$ is a lower bound for the real roots of the function.

Lower Bound: $- 4$

Step 10

Apply synthetic division on $\frac{7 x^{4} + 7 x^{3} - x^{2} - 28 x - 28}{x - 7}$ when $x = 7$ .

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

$7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0

The first number in the dividend $(7)$ is put into the first position of the result area (below the horizontal line).

$7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0
	$7$

Multiply the newest entry in the result $(7)$ by the divisor $(7)$ and place the result of $(49)$ under the next term in the dividend $(7)$ .

$7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$49$			\xa0
	$7$

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

$7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$49$			\xa0
	$7$	$56$

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 $(- 1)$ .

$7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$49$	$392$		\xa0
	$7$	$56$

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

$7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$49$	$392$		\xa0
	$7$	$56$	$391$

Multiply the newest entry in the result $(391)$ by the divisor $(7)$ and place the result of $(2737)$ under the next term in the dividend $(- 28)$ .

$7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$49$	$392$	$2737$	\xa0
	$7$	$56$	$391$

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

$7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$49$	$392$	$2737$	\xa0
	$7$	$56$	$391$	$2709$

Multiply the newest entry in the result $(2709)$ by the divisor $(7)$ and place the result of $(18963)$ under the next term in the dividend $(- 28)$ .

$7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$49$	$392$	$2737$	$18963$
	$7$	$56$	$391$	$2709$

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

$7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$49$	$392$	$2737$	$18963$
	$7$	$56$	$391$	$2709$	$18935$

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

$7 x^{3} + 56 x^{2} + (391) x + 2709 + \frac{18935}{x - 7}$

Simplify the quotient polynomial.

$7 x^{3} + 56 x^{2} + 391 x + 2709 + \frac{18935}{x - 7}$

Step 11

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 12

Apply synthetic division on $\frac{7 x^{4} + 7 x^{3} - x^{2} - 28 x - 28}{x + 7}$ when $x = - 7$ .

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

$- 7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0

The first number in the dividend $(7)$ is put into the first position of the result area (below the horizontal line).

$- 7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0
	$7$

Multiply the newest entry in the result $(7)$ by the divisor $(- 7)$ and place the result of $(- 49)$ under the next term in the dividend $(7)$ .

$- 7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 49$			\xa0
	$7$

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

$- 7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 49$			\xa0
	$7$	$- 42$

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 $(- 1)$ .

$- 7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 49$	$294$		\xa0
	$7$	$- 42$

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

$- 7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 49$	$294$		\xa0
	$7$	$- 42$	$293$

Multiply the newest entry in the result $(293)$ by the divisor $(- 7)$ and place the result of $(- 2051)$ under the next term in the dividend $(- 28)$ .

$- 7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 49$	$294$	$- 2051$	\xa0
	$7$	$- 42$	$293$

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

$- 7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 49$	$294$	$- 2051$	\xa0
	$7$	$- 42$	$293$	$- 2079$

Multiply the newest entry in the result $(- 2079)$ by the divisor $(- 7)$ and place the result of $(14553)$ under the next term in the dividend $(- 28)$ .

$- 7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 49$	$294$	$- 2051$	$14553$
	$7$	$- 42$	$293$	$- 2079$

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

$- 7$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 49$	$294$	$- 2051$	$14553$
	$7$	$- 42$	$293$	$- 2079$	$14525$

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

$7 x^{3} + - 42 x^{2} + (293) x - 2079 + \frac{14525}{x + 7}$

Simplify the quotient polynomial.

$7 x^{3} - 42 x^{2} + 293 x - 2079 + \frac{14525}{x + 7}$

Step 13

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 14

Apply synthetic division on $\frac{7 x^{4} + 7 x^{3} - x^{2} - 28 x - 28}{x - 14}$ when $x = 14$ .

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

$14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0

The first number in the dividend $(7)$ is put into the first position of the result area (below the horizontal line).

$14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0
	$7$

Multiply the newest entry in the result $(7)$ by the divisor $(14)$ and place the result of $(98)$ under the next term in the dividend $(7)$ .

$14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$98$			\xa0
	$7$

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

$14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$98$			\xa0
	$7$	$105$

Multiply the newest entry in the result $(105)$ by the divisor $(14)$ and place the result of $(1470)$ under the next term in the dividend $(- 1)$ .

$14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$98$	$1470$		\xa0
	$7$	$105$

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

$14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$98$	$1470$		\xa0
	$7$	$105$	$1469$

Multiply the newest entry in the result $(1469)$ by the divisor $(14)$ and place the result of $(20566)$ under the next term in the dividend $(- 28)$ .

$14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$98$	$1470$	$20566$	\xa0
	$7$	$105$	$1469$

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

$14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$98$	$1470$	$20566$	\xa0
	$7$	$105$	$1469$	$20538$

Multiply the newest entry in the result $(20538)$ by the divisor $(14)$ and place the result of $(287532)$ under the next term in the dividend $(- 28)$ .

$14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$98$	$1470$	$20566$	$287532$
	$7$	$105$	$1469$	$20538$

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

$14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$98$	$1470$	$20566$	$287532$
	$7$	$105$	$1469$	$20538$	$287504$

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

$7 x^{3} + 105 x^{2} + (1469) x + 20538 + \frac{287504}{x - 14}$

Simplify the quotient polynomial.

$7 x^{3} + 105 x^{2} + 1469 x + 20538 + \frac{287504}{x - 14}$

Step 15

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: $14$

Step 16

Apply synthetic division on $\frac{7 x^{4} + 7 x^{3} - x^{2} - 28 x - 28}{x + 14}$ when $x = - 14$ .

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

$- 14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0

The first number in the dividend $(7)$ is put into the first position of the result area (below the horizontal line).

$- 14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0
	$7$

Multiply the newest entry in the result $(7)$ by the divisor $(- 14)$ and place the result of $(- 98)$ under the next term in the dividend $(7)$ .

$- 14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 98$			\xa0
	$7$

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

$- 14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 98$			\xa0
	$7$	$- 91$

Multiply the newest entry in the result $(- 91)$ by the divisor $(- 14)$ and place the result of $(1274)$ under the next term in the dividend $(- 1)$ .

$- 14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 98$	$1274$		\xa0
	$7$	$- 91$

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

$- 14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 98$	$1274$		\xa0
	$7$	$- 91$	$1273$

Multiply the newest entry in the result $(1273)$ by the divisor $(- 14)$ and place the result of $(- 17822)$ under the next term in the dividend $(- 28)$ .

$- 14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 98$	$1274$	$- 17822$	\xa0
	$7$	$- 91$	$1273$

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

$- 14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 98$	$1274$	$- 17822$	\xa0
	$7$	$- 91$	$1273$	$- 17850$

Multiply the newest entry in the result $(- 17850)$ by the divisor $(- 14)$ and place the result of $(249900)$ under the next term in the dividend $(- 28)$ .

$- 14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 98$	$1274$	$- 17822$	$249900$
	$7$	$- 91$	$1273$	$- 17850$

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

$- 14$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 98$	$1274$	$- 17822$	$249900$
	$7$	$- 91$	$1273$	$- 17850$	$249872$

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

$7 x^{3} + - 91 x^{2} + (1273) x - 17850 + \frac{249872}{x + 14}$

Simplify the quotient polynomial.

$7 x^{3} - 91 x^{2} + 1273 x - 17850 + \frac{249872}{x + 14}$

Step 17

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: $- 14$

Step 18

Apply synthetic division on $\frac{7 x^{4} + 7 x^{3} - x^{2} - 28 x - 28}{x - 28}$ when $x = 28$ .

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

$28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0

The first number in the dividend $(7)$ is put into the first position of the result area (below the horizontal line).

$28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0
	$7$

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

$28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$196$			\xa0
	$7$

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

$28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$196$			\xa0
	$7$	$203$

Multiply the newest entry in the result $(203)$ by the divisor $(28)$ and place the result of $(5684)$ under the next term in the dividend $(- 1)$ .

$28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$196$	$5684$		\xa0
	$7$	$203$

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

$28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$196$	$5684$		\xa0
	$7$	$203$	$5683$

Multiply the newest entry in the result $(5683)$ by the divisor $(28)$ and place the result of $(159124)$ under the next term in the dividend $(- 28)$ .

$28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$196$	$5684$	$159124$	\xa0
	$7$	$203$	$5683$

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

$28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$196$	$5684$	$159124$	\xa0
	$7$	$203$	$5683$	$159096$

Multiply the newest entry in the result $(159096)$ by the divisor $(28)$ and place the result of $(4454688)$ under the next term in the dividend $(- 28)$ .

$28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$196$	$5684$	$159124$	$4454688$
	$7$	$203$	$5683$	$159096$

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

$28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$196$	$5684$	$159124$	$4454688$
	$7$	$203$	$5683$	$159096$	$4454660$

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

$7 x^{3} + 203 x^{2} + (5683) x + 159096 + \frac{4454660}{x - 28}$

Simplify the quotient polynomial.

$7 x^{3} + 203 x^{2} + 5683 x + 159096 + \frac{4454660}{x - 28}$

Step 19

Since $28 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $28$ is an upper bound for the real roots of the function.

Upper Bound: $28$

Step 20

Apply synthetic division on $\frac{7 x^{4} + 7 x^{3} - x^{2} - 28 x - 28}{x + 28}$ when $x = - 28$ .

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

$- 28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0

The first number in the dividend $(7)$ is put into the first position of the result area (below the horizontal line).

$- 28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0				\xa0
	$7$

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

$- 28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 196$			\xa0
	$7$

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

$- 28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 196$			\xa0
	$7$	$- 189$

Multiply the newest entry in the result $(- 189)$ by the divisor $(- 28)$ and place the result of $(5292)$ under the next term in the dividend $(- 1)$ .

$- 28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 196$	$5292$		\xa0
	$7$	$- 189$

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

$- 28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 196$	$5292$		\xa0
	$7$	$- 189$	$5291$

Multiply the newest entry in the result $(5291)$ by the divisor $(- 28)$ and place the result of $(- 148148)$ under the next term in the dividend $(- 28)$ .

$- 28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 196$	$5292$	$- 148148$	\xa0
	$7$	$- 189$	$5291$

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

$- 28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 196$	$5292$	$- 148148$	\xa0
	$7$	$- 189$	$5291$	$- 148176$

Multiply the newest entry in the result $(- 148176)$ by the divisor $(- 28)$ and place the result of $(4148928)$ under the next term in the dividend $(- 28)$ .

$- 28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 196$	$5292$	$- 148148$	$4148928$
	$7$	$- 189$	$5291$	$- 148176$

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

$- 28$	$7$	$7$	$- 1$	$- 28$	$- 28$
	\xa0	$- 196$	$5292$	$- 148148$	$4148928$
	$7$	$- 189$	$5291$	$- 148176$	$4148900$

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

$7 x^{3} + - 189 x^{2} + (5291) x - 148176 + \frac{4148900}{x + 28}$

Simplify the quotient polynomial.

$7 x^{3} - 189 x^{2} + 5291 x - 148176 + \frac{4148900}{x + 28}$

Step 21

Since $- 28 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 28$ is a lower bound for the real roots of the function.

Lower Bound: $- 28$

Step 22

Determine the upper and lower bounds.

Upper Bounds: $2, 4, 7, 14, 28$

Lower Bounds: $- 2, - 4, - 7, - 14, - 28$

Step 23

Find the Upper and Lower Bounds 7x^4+7x^3-x^2-28x-28

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