Find the Upper and Lower Bounds x^4+x^3-7x^2-5x+10

$x^{4} + x^{3} - 7 x^{2} - 5 x + 10$

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.

$p = \pm 1, \pm 2, \pm 5, \pm 10$

$q = \pm 1$

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

$\pm 1, \pm 2, \pm 5, \pm 10$

Step 2

Apply synthetic division on $\frac{x^{4} + x^{3} - 7 x^{2} - 5 x + 10}{x - 5}$ when $x = 5$ .

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

$5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0				\xa0

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

$5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(5)$ and place the result of $(5)$ under the next term in the dividend $(1)$ .

$5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$5$			\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.

$5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$5$			\xa0
	$1$	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(5)$ and place the result of $(30)$ under the next term in the dividend $(- 7)$ .

$5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$5$	$30$		\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.

$5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$5$	$30$		\xa0
	$1$	$6$	$23$

Multiply the newest entry in the result $(23)$ by the divisor $(5)$ and place the result of $(115)$ under the next term in the dividend $(- 5)$ .

$5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$5$	$30$	$115$	\xa0
	$1$	$6$	$23$

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

$5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$5$	$30$	$115$	\xa0
	$1$	$6$	$23$	$110$

Multiply the newest entry in the result $(110)$ by the divisor $(5)$ and place the result of $(550)$ under the next term in the dividend $(10)$ .

$5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$5$	$30$	$115$	$550$
	$1$	$6$	$23$	$110$

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

$5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$5$	$30$	$115$	$550$
	$1$	$6$	$23$	$110$	$560$

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

$1 x^{3} + 6 x^{2} + (23) x + 110 + \frac{560}{x - 5}$

Simplify the quotient polynomial.

$x^{3} + 6 x^{2} + 23 x + 110 + \frac{560}{x - 5}$

Step 3

Since $5 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $5$ is an upper bound for the real roots of the function.

Upper Bound: $5$

Step 4

Apply synthetic division on $\frac{x^{4} + x^{3} - 7 x^{2} - 5 x + 10}{x + 5}$ when $x = - 5$ .

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

$- 5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0				\xa0

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

$- 5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 5)$ and place the result of $(- 5)$ under the next term in the dividend $(1)$ .

$- 5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 5$			\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.

$- 5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 5$			\xa0
	$1$	$- 4$

Multiply the newest entry in the result $(- 4)$ by the divisor $(- 5)$ and place the result of $(20)$ under the next term in the dividend $(- 7)$ .

$- 5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 5$	$20$		\xa0
	$1$	$- 4$

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

$- 5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 5$	$20$		\xa0
	$1$	$- 4$	$13$

Multiply the newest entry in the result $(13)$ by the divisor $(- 5)$ and place the result of $(- 65)$ under the next term in the dividend $(- 5)$ .

$- 5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 5$	$20$	$- 65$	\xa0
	$1$	$- 4$	$13$

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

$- 5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 5$	$20$	$- 65$	\xa0
	$1$	$- 4$	$13$	$- 70$

Multiply the newest entry in the result $(- 70)$ by the divisor $(- 5)$ and place the result of $(350)$ under the next term in the dividend $(10)$ .

$- 5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 5$	$20$	$- 65$	$350$
	$1$	$- 4$	$13$	$- 70$

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

$- 5$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 5$	$20$	$- 65$	$350$
	$1$	$- 4$	$13$	$- 70$	$360$

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

$1 x^{3} + - 4 x^{2} + (13) x - 70 + \frac{360}{x + 5}$

Simplify the quotient polynomial.

$x^{3} - 4 x^{2} + 13 x - 70 + \frac{360}{x + 5}$

Step 5

Since $- 5 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 5$ is a lower bound for the real roots of the function.

Lower Bound: $- 5$

Step 6

Apply synthetic division on $\frac{x^{4} + x^{3} - 7 x^{2} - 5 x + 10}{x - 10}$ when $x = 10$ .

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

$10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0				\xa0

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

$10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0				\xa0
	$1$

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

$10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$10$			\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.

$10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$10$			\xa0
	$1$	$11$

Multiply the newest entry in the result $(11)$ by the divisor $(10)$ and place the result of $(110)$ under the next term in the dividend $(- 7)$ .

$10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$10$	$110$		\xa0
	$1$	$11$

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

$10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$10$	$110$		\xa0
	$1$	$11$	$103$

Multiply the newest entry in the result $(103)$ by the divisor $(10)$ and place the result of $(1030)$ under the next term in the dividend $(- 5)$ .

$10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$10$	$110$	$1030$	\xa0
	$1$	$11$	$103$

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

$10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$10$	$110$	$1030$	\xa0
	$1$	$11$	$103$	$1025$

Multiply the newest entry in the result $(1025)$ by the divisor $(10)$ and place the result of $(10250)$ under the next term in the dividend $(10)$ .

$10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$10$	$110$	$1030$	$10250$
	$1$	$11$	$103$	$1025$

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

$10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$10$	$110$	$1030$	$10250$
	$1$	$11$	$103$	$1025$	$10260$

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

$1 x^{3} + 11 x^{2} + (103) x + 1025 + \frac{10260}{x - 10}$

Simplify the quotient polynomial.

$x^{3} + 11 x^{2} + 103 x + 1025 + \frac{10260}{x - 10}$

Step 7

Since $10 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $10$ is an upper bound for the real roots of the function.

Upper Bound: $10$

Step 8

Apply synthetic division on $\frac{x^{4} + x^{3} - 7 x^{2} - 5 x + 10}{x + 10}$ when $x = - 10$ .

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

$- 10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0				\xa0

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

$- 10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0				\xa0
	$1$

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

$- 10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 10$			\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.

$- 10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 10$			\xa0
	$1$	$- 9$

Multiply the newest entry in the result $(- 9)$ by the divisor $(- 10)$ and place the result of $(90)$ under the next term in the dividend $(- 7)$ .

$- 10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 10$	$90$		\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.

$- 10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 10$	$90$		\xa0
	$1$	$- 9$	$83$

Multiply the newest entry in the result $(83)$ by the divisor $(- 10)$ and place the result of $(- 830)$ under the next term in the dividend $(- 5)$ .

$- 10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 10$	$90$	$- 830$	\xa0
	$1$	$- 9$	$83$

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

$- 10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 10$	$90$	$- 830$	\xa0
	$1$	$- 9$	$83$	$- 835$

Multiply the newest entry in the result $(- 835)$ by the divisor $(- 10)$ and place the result of $(8350)$ under the next term in the dividend $(10)$ .

$- 10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 10$	$90$	$- 830$	$8350$
	$1$	$- 9$	$83$	$- 835$

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

$- 10$	$1$	$1$	$- 7$	$- 5$	$10$
	\xa0	$- 10$	$90$	$- 830$	$8350$
	$1$	$- 9$	$83$	$- 835$	$8360$

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

$1 x^{3} + - 9 x^{2} + (83) x - 835 + \frac{8360}{x + 10}$

Simplify the quotient polynomial.

$x^{3} - 9 x^{2} + 83 x - 835 + \frac{8360}{x + 10}$

Step 9

Since $- 10 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 10$ is a lower bound for the real roots of the function.

Lower Bound: $- 10$

Step 10

Determine the upper and lower bounds.

Upper Bounds: $5, 10$

Lower Bounds: $- 5, - 10$

Step 11

Find the Upper and Lower Bounds x^4+x^3-7x^2-5x+10

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