Find the Upper and Lower Bounds 4x^4-16x^3+11x^2+4x-3

$4 x^{4} - 16 x^{3} + 11 x^{2} + 4 x - 3$

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.

$p = \pm 1, \pm 3$

$q = \pm 1, \pm 2, \pm 4$

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}$

Step 2

Apply synthetic division on $\frac{4 x^{4} - 16 x^{3} + 11 x^{2} + 4 x - 3}{x + 1}$ when $x = - 1$ .

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

$- 1$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0				\xa0

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0				\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(- 1)$ and place the result of $(- 4)$ under the next term in the dividend $(- 16)$ .

$- 1$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 4$			\xa0
	$4$

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

$- 1$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 4$			\xa0
	$4$	$- 20$

Multiply the newest entry in the result $(- 20)$ by the divisor $(- 1)$ and place the result of $(20)$ under the next term in the dividend $(11)$ .

$- 1$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 4$	$20$		\xa0
	$4$	$- 20$

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

$- 1$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 4$	$20$		\xa0
	$4$	$- 20$	$31$

Multiply the newest entry in the result $(31)$ by the divisor $(- 1)$ and place the result of $(- 31)$ under the next term in the dividend $(4)$ .

$- 1$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 4$	$20$	$- 31$	\xa0
	$4$	$- 20$	$31$

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

$- 1$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 4$	$20$	$- 31$	\xa0
	$4$	$- 20$	$31$	$- 27$

Multiply the newest entry in the result $(- 27)$ by the divisor $(- 1)$ and place the result of $(27)$ under the next term in the dividend $(- 3)$ .

$- 1$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 4$	$20$	$- 31$	$27$
	$4$	$- 20$	$31$	$- 27$

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

$- 1$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 4$	$20$	$- 31$	$27$
	$4$	$- 20$	$31$	$- 27$	$24$

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

$4 x^{3} + - 20 x^{2} + (31) x - 27 + \frac{24}{x + 1}$

Simplify the quotient polynomial.

$4 x^{3} - 20 x^{2} + 31 x - 27 + \frac{24}{x + 1}$

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 $\frac{4 x^{4} - 16 x^{3} + 11 x^{2} + 4 x - 3}{x + \frac{1}{2}}$ when $x = - \frac{1}{2}$ .

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

$- \frac{1}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0				\xa0

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- \frac{1}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0				\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 2)$ under the next term in the dividend $(- 16)$ .

$- \frac{1}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 2$			\xa0
	$4$

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

$- \frac{1}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 2$			\xa0
	$4$	$- 18$

Multiply the newest entry in the result $(- 18)$ by the divisor $(- \frac{1}{2})$ and place the result of $(9)$ under the next term in the dividend $(11)$ .

$- \frac{1}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 2$	$9$		\xa0
	$4$	$- 18$

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

$- \frac{1}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 2$	$9$		\xa0
	$4$	$- 18$	$20$

Multiply the newest entry in the result $(20)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 10)$ under the next term in the dividend $(4)$ .

$- \frac{1}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 2$	$9$	$- 10$	\xa0
	$4$	$- 18$	$20$

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

$- \frac{1}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 2$	$9$	$- 10$	\xa0
	$4$	$- 18$	$20$	$- 6$

Multiply the newest entry in the result $(- 6)$ by the divisor $(- \frac{1}{2})$ and place the result of $(3)$ under the next term in the dividend $(- 3)$ .

$- \frac{1}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 2$	$9$	$- 10$	$3$
	$4$	$- 18$	$20$	$- 6$

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

$- \frac{1}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 2$	$9$	$- 10$	$3$
	$4$	$- 18$	$20$	$- 6$	$0$

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

$4 x^{3} + - 18 x^{2} + (20) x - 6$

Simplify the quotient polynomial.

$4 x^{3} - 18 x^{2} + 20 x - 6$

Step 5

Since $- \frac{1}{2} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{1}{2}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{1}{2}$

Step 6

Apply synthetic division on $\frac{4 x^{4} - 16 x^{3} + 11 x^{2} + 4 x - 3}{x + 3}$ when $x = - 3$ .

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

$- 3$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0				\xa0

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- 3$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0				\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(- 3)$ and place the result of $(- 12)$ under the next term in the dividend $(- 16)$ .

$- 3$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 12$			\xa0
	$4$

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

$- 3$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 12$			\xa0
	$4$	$- 28$

Multiply the newest entry in the result $(- 28)$ by the divisor $(- 3)$ and place the result of $(84)$ under the next term in the dividend $(11)$ .

$- 3$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 12$	$84$		\xa0
	$4$	$- 28$

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

$- 3$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 12$	$84$		\xa0
	$4$	$- 28$	$95$

Multiply the newest entry in the result $(95)$ by the divisor $(- 3)$ and place the result of $(- 285)$ under the next term in the dividend $(4)$ .

$- 3$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 12$	$84$	$- 285$	\xa0
	$4$	$- 28$	$95$

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

$- 3$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 12$	$84$	$- 285$	\xa0
	$4$	$- 28$	$95$	$- 281$

Multiply the newest entry in the result $(- 281)$ by the divisor $(- 3)$ and place the result of $(843)$ under the next term in the dividend $(- 3)$ .

$- 3$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 12$	$84$	$- 285$	$843$
	$4$	$- 28$	$95$	$- 281$

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

$- 3$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 12$	$84$	$- 285$	$843$
	$4$	$- 28$	$95$	$- 281$	$840$

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

$4 x^{3} + - 28 x^{2} + (95) x - 281 + \frac{840}{x + 3}$

Simplify the quotient polynomial.

$4 x^{3} - 28 x^{2} + 95 x - 281 + \frac{840}{x + 3}$

Step 7

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: $- 3$

Step 8

Apply synthetic division on $\frac{4 x^{4} - 16 x^{3} + 11 x^{2} + 4 x - 3}{x + \frac{3}{2}}$ when $x = - \frac{3}{2}$ .

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

$- \frac{3}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0				\xa0

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- \frac{3}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0				\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(- \frac{3}{2})$ and place the result of $(- 6)$ under the next term in the dividend $(- 16)$ .

$- \frac{3}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 6$			\xa0
	$4$

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

$- \frac{3}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 6$			\xa0
	$4$	$- 22$

Multiply the newest entry in the result $(- 22)$ by the divisor $(- \frac{3}{2})$ and place the result of $(33)$ under the next term in the dividend $(11)$ .

$- \frac{3}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 6$	$33$		\xa0
	$4$	$- 22$

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

$- \frac{3}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 6$	$33$		\xa0
	$4$	$- 22$	$44$

Multiply the newest entry in the result $(44)$ by the divisor $(- \frac{3}{2})$ and place the result of $(- 66)$ under the next term in the dividend $(4)$ .

$- \frac{3}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 6$	$33$	$- 66$	\xa0
	$4$	$- 22$	$44$

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

$- \frac{3}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 6$	$33$	$- 66$	\xa0
	$4$	$- 22$	$44$	$- 62$

Multiply the newest entry in the result $(- 62)$ by the divisor $(- \frac{3}{2})$ and place the result of $(93)$ under the next term in the dividend $(- 3)$ .

$- \frac{3}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 6$	$33$	$- 66$	$93$
	$4$	$- 22$	$44$	$- 62$

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

$- \frac{3}{2}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 6$	$33$	$- 66$	$93$
	$4$	$- 22$	$44$	$- 62$	$90$

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

$4 x^{3} + - 22 x^{2} + (44) x - 62 + \frac{90}{x + \frac{3}{2}}$

Simplify the quotient polynomial.

$4 x^{3} - 22 x^{2} + 44 x - 62 + \frac{180}{2 x + 3}$

Step 9

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

Step 10

Apply synthetic division on $\frac{4 x^{4} - 16 x^{3} + 11 x^{2} + 4 x - 3}{x + \frac{3}{4}}$ when $x = - \frac{3}{4}$ .

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

$- \frac{3}{4}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0				\xa0

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- \frac{3}{4}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0				\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(- \frac{3}{4})$ and place the result of $(- 3)$ under the next term in the dividend $(- 16)$ .

$- \frac{3}{4}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 3$			\xa0
	$4$

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

$- \frac{3}{4}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 3$			\xa0
	$4$	$- 19$

Multiply the newest entry in the result $(- 19)$ by the divisor $(- \frac{3}{4})$ and place the result of $(\frac{57}{4})$ under the next term in the dividend $(11)$ .

$- \frac{3}{4}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 3$	$\frac{57}{4}$		\xa0
	$4$	$- 19$

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

$- \frac{3}{4}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 3$	$\frac{57}{4}$		\xa0
	$4$	$- 19$	$\frac{101}{4}$

Multiply the newest entry in the result $(\frac{101}{4})$ by the divisor $(- \frac{3}{4})$ and place the result of $(- \frac{303}{16})$ under the next term in the dividend $(4)$ .

$- \frac{3}{4}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 3$	$\frac{57}{4}$	$- \frac{303}{16}$	\xa0
	$4$	$- 19$	$\frac{101}{4}$

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

$- \frac{3}{4}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 3$	$\frac{57}{4}$	$- \frac{303}{16}$	\xa0
	$4$	$- 19$	$\frac{101}{4}$	$- \frac{239}{16}$

Multiply the newest entry in the result $(- \frac{239}{16})$ by the divisor $(- \frac{3}{4})$ and place the result of $(\frac{717}{64})$ under the next term in the dividend $(- 3)$ .

$- \frac{3}{4}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 3$	$\frac{57}{4}$	$- \frac{303}{16}$	$\frac{717}{64}$
	$4$	$- 19$	$\frac{101}{4}$	$- \frac{239}{16}$

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

$- \frac{3}{4}$	$4$	$- 16$	$11$	$4$	$- 3$
	\xa0	$- 3$	$\frac{57}{4}$	$- \frac{303}{16}$	$\frac{717}{64}$
	$4$	$- 19$	$\frac{101}{4}$	$- \frac{239}{16}$	$\frac{525}{64}$

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

$4 x^{3} + - 19 x^{2} + (\frac{101}{4}) x - \frac{239}{16} + \frac{\frac{525}{64}}{x + \frac{3}{4}}$

Simplify the quotient polynomial.

$4 x^{3} - 19 x^{2} + \frac{101 x}{4} - \frac{239}{16} + \frac{525}{16 (4 x + 3)}$

Step 11

Since $- \frac{3}{4} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{3}{4}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{3}{4}$

Step 12

Determine the upper and lower bounds.

No Upper Bounds

Lower Bounds: $- 1, - \frac{1}{2}, - 3, - \frac{3}{2}, - \frac{3}{4}$

Step 13

Find the Upper and Lower Bounds 4x^4-16x^3+11x^2+4x-3

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