Finite Math Examples

$f (x) = - x^{2} + 6 x^{2} - 9 x + 6$

Step 1

Add $- x^{2}$ and $6 x^{2}$ .

$f (x) = 5 x^{2} - 9 x + 6$

Step 2

Find every combination of $\pm \frac{p}{q}$ .

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Step 2.1

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.

$p = \pm 1, \pm 2, \pm 3, \pm 6$

$q = \pm 1, \pm 5$

Step 2.2

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

$\pm 1, \pm \frac{1}{5}, \pm 2, \pm \frac{2}{5}, \pm 3, \pm \frac{3}{5}, \pm 6, \pm \frac{6}{5}$

Step 3

Apply synthetic division on $\frac{5 x^{2} - 9 x + 6}{x + 1}$ when $x = - 1$ .

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Step 3.1

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

$- 1$	$5$	$- 9$	$6$

Step 3.2

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$5$	$- 9$	$6$

	$5$

Step 3.3

Multiply the newest entry in the result $(5)$ by the divisor $(- 1)$ and place the result of $(- 5)$ under the next term in the dividend $(- 9)$ .

$- 1$	$5$	$- 9$	$6$
		$- 5$
	$5$

Step 3.4

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

$- 1$	$5$	$- 9$	$6$
		$- 5$
	$5$	$- 14$

Step 3.5

Multiply the newest entry in the result $(- 14)$ by the divisor $(- 1)$ and place the result of $(14)$ under the next term in the dividend $(6)$ .

$- 1$	$5$	$- 9$	$6$
		$- 5$	$14$
	$5$	$- 14$

Step 3.6

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

$- 1$	$5$	$- 9$	$6$
		$- 5$	$14$
	$5$	$- 14$	$20$

Step 3.7

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

$(5) x - 14 + \frac{20}{x + 1}$

Step 3.8

Simplify the quotient polynomial.

$5 x - 14 + \frac{20}{x + 1}$

Step 4

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 5

Apply synthetic division on $\frac{5 x^{2} - 9 x + 6}{x + \frac{1}{5}}$ when $x = - \frac{1}{5}$ .

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Step 5.1

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

$- \frac{1}{5}$	$5$	$- 9$	$6$

Step 5.2

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$- \frac{1}{5}$	$5$	$- 9$	$6$

	$5$

Step 5.3

Multiply the newest entry in the result $(5)$ by the divisor $(- \frac{1}{5})$ and place the result of $(- 1)$ under the next term in the dividend $(- 9)$ .

$- \frac{1}{5}$	$5$	$- 9$	$6$
		$- 1$
	$5$

Step 5.4

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

$- \frac{1}{5}$	$5$	$- 9$	$6$
		$- 1$
	$5$	$- 10$

Step 5.5

Multiply the newest entry in the result $(- 10)$ by the divisor $(- \frac{1}{5})$ and place the result of $(2)$ under the next term in the dividend $(6)$ .

$- \frac{1}{5}$	$5$	$- 9$	$6$
		$- 1$	$2$
	$5$	$- 10$

Step 5.6

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

$- \frac{1}{5}$	$5$	$- 9$	$6$
		$- 1$	$2$
	$5$	$- 10$	$8$

Step 5.7

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

$(5) x - 10 + \frac{8}{x + \frac{1}{5}}$

Step 5.8

Simplify the quotient polynomial.

$5 x - 10 + \frac{40}{5 x + 1}$

Step 6

Since $- \frac{1}{5} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{1}{5}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{1}{5}$

Step 7

Apply synthetic division on $\frac{5 x^{2} - 9 x + 6}{x - 2}$ when $x = 2$ .

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Step 7.1

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

$2$	$5$	$- 9$	$6$

Step 7.2

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$2$	$5$	$- 9$	$6$

	$5$

Step 7.3

Multiply the newest entry in the result $(5)$ by the divisor $(2)$ and place the result of $(10)$ under the next term in the dividend $(- 9)$ .

$2$	$5$	$- 9$	$6$
		$10$
	$5$

Step 7.4

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

$2$	$5$	$- 9$	$6$
		$10$
	$5$	$1$

Step 7.5

Multiply the newest entry in the result $(1)$ by the divisor $(2)$ and place the result of $(2)$ under the next term in the dividend $(6)$ .

$2$	$5$	$- 9$	$6$
		$10$	$2$
	$5$	$1$

Step 7.6

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

$2$	$5$	$- 9$	$6$
		$10$	$2$
	$5$	$1$	$8$

Step 7.7

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

$(5) x + 1 + \frac{8}{x - 2}$

Step 7.8

Simplify the quotient polynomial.

$5 x + 1 + \frac{8}{x - 2}$

Step 8

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 9

Apply synthetic division on $\frac{5 x^{2} - 9 x + 6}{x + 2}$ when $x = - 2$ .

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Step 9.1

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

$- 2$	$5$	$- 9$	$6$

Step 9.2

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$- 2$	$5$	$- 9$	$6$

	$5$

Step 9.3

Multiply the newest entry in the result $(5)$ by the divisor $(- 2)$ and place the result of $(- 10)$ under the next term in the dividend $(- 9)$ .

$- 2$	$5$	$- 9$	$6$
		$- 10$
	$5$

Step 9.4

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

$- 2$	$5$	$- 9$	$6$
		$- 10$
	$5$	$- 19$

Step 9.5

Multiply the newest entry in the result $(- 19)$ by the divisor $(- 2)$ and place the result of $(38)$ under the next term in the dividend $(6)$ .

$- 2$	$5$	$- 9$	$6$
		$- 10$	$38$
	$5$	$- 19$

Step 9.6

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

$- 2$	$5$	$- 9$	$6$
		$- 10$	$38$
	$5$	$- 19$	$44$

Step 9.7

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

$(5) x - 19 + \frac{44}{x + 2}$

Step 9.8

Simplify the quotient polynomial.

$5 x - 19 + \frac{44}{x + 2}$

Step 10

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 11

Apply synthetic division on $\frac{5 x^{2} - 9 x + 6}{x + \frac{2}{5}}$ when $x = - \frac{2}{5}$ .

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Step 11.1

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

$- \frac{2}{5}$	$5$	$- 9$	$6$

Step 11.2

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$- \frac{2}{5}$	$5$	$- 9$	$6$

	$5$

Step 11.3

Multiply the newest entry in the result $(5)$ by the divisor $(- \frac{2}{5})$ and place the result of $(- 2)$ under the next term in the dividend $(- 9)$ .

$- \frac{2}{5}$	$5$	$- 9$	$6$
		$- 2$
	$5$

Step 11.4

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

$- \frac{2}{5}$	$5$	$- 9$	$6$
		$- 2$
	$5$	$- 11$

Step 11.5

Multiply the newest entry in the result $(- 11)$ by the divisor $(- \frac{2}{5})$ and place the result of $(\frac{22}{5})$ under the next term in the dividend $(6)$ .

$- \frac{2}{5}$	$5$	$- 9$	$6$
		$- 2$	$\frac{22}{5}$
	$5$	$- 11$

Step 11.6

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

$- \frac{2}{5}$	$5$	$- 9$	$6$
		$- 2$	$\frac{22}{5}$
	$5$	$- 11$	$\frac{52}{5}$

Step 11.7

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

$(5) x - 11 + \frac{\frac{52}{5}}{x + \frac{2}{5}}$

Step 11.8

Simplify the quotient polynomial.

$5 x - 11 + \frac{52}{5 x + 2}$

Step 12

Since $- \frac{2}{5} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{2}{5}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{2}{5}$

Step 13

Apply synthetic division on $\frac{5 x^{2} - 9 x + 6}{x - 3}$ when $x = 3$ .

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Step 13.1

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

$3$	$5$	$- 9$	$6$

Step 13.2

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$3$	$5$	$- 9$	$6$

	$5$

Step 13.3

Multiply the newest entry in the result $(5)$ by the divisor $(3)$ and place the result of $(15)$ under the next term in the dividend $(- 9)$ .

$3$	$5$	$- 9$	$6$
		$15$
	$5$

Step 13.4

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

$3$	$5$	$- 9$	$6$
		$15$
	$5$	$6$

Step 13.5

Multiply the newest entry in the result $(6)$ by the divisor $(3)$ and place the result of $(18)$ under the next term in the dividend $(6)$ .

$3$	$5$	$- 9$	$6$
		$15$	$18$
	$5$	$6$

Step 13.6

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

$3$	$5$	$- 9$	$6$
		$15$	$18$
	$5$	$6$	$24$

Step 13.7

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

$(5) x + 6 + \frac{24}{x - 3}$

Step 13.8

Simplify the quotient polynomial.

$5 x + 6 + \frac{24}{x - 3}$

Step 14

Since $3 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $3$ is an upper bound for the real roots of the function.

Upper Bound: $3$

Step 15

Apply synthetic division on $\frac{5 x^{2} - 9 x + 6}{x + 3}$ when $x = - 3$ .

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Step 15.1

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

$- 3$	$5$	$- 9$	$6$

Step 15.2

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$- 3$	$5$	$- 9$	$6$

	$5$

Step 15.3

Multiply the newest entry in the result $(5)$ by the divisor $(- 3)$ and place the result of $(- 15)$ under the next term in the dividend $(- 9)$ .

$- 3$	$5$	$- 9$	$6$
		$- 15$
	$5$

Step 15.4

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

$- 3$	$5$	$- 9$	$6$
		$- 15$
	$5$	$- 24$

Step 15.5

Multiply the newest entry in the result $(- 24)$ by the divisor $(- 3)$ and place the result of $(72)$ under the next term in the dividend $(6)$ .

$- 3$	$5$	$- 9$	$6$
		$- 15$	$72$
	$5$	$- 24$

Step 15.6

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

$- 3$	$5$	$- 9$	$6$
		$- 15$	$72$
	$5$	$- 24$	$78$

Step 15.7

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

$(5) x - 24 + \frac{78}{x + 3}$

Step 15.8

Simplify the quotient polynomial.

$5 x - 24 + \frac{78}{x + 3}$

Step 16

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 17

Apply synthetic division on $\frac{5 x^{2} - 9 x + 6}{x + \frac{3}{5}}$ when $x = - \frac{3}{5}$ .

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Step 17.1

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

$- \frac{3}{5}$	$5$	$- 9$	$6$

Step 17.2

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$- \frac{3}{5}$	$5$	$- 9$	$6$

	$5$

Step 17.3

Multiply the newest entry in the result $(5)$ by the divisor $(- \frac{3}{5})$ and place the result of $(- 3)$ under the next term in the dividend $(- 9)$ .

$- \frac{3}{5}$	$5$	$- 9$	$6$
		$- 3$
	$5$

Step 17.4

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

$- \frac{3}{5}$	$5$	$- 9$	$6$
		$- 3$
	$5$	$- 12$

Step 17.5

Multiply the newest entry in the result $(- 12)$ by the divisor $(- \frac{3}{5})$ and place the result of $(\frac{36}{5})$ under the next term in the dividend $(6)$ .

$- \frac{3}{5}$	$5$	$- 9$	$6$
		$- 3$	$\frac{36}{5}$
	$5$	$- 12$

Step 17.6

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

$- \frac{3}{5}$	$5$	$- 9$	$6$
		$- 3$	$\frac{36}{5}$
	$5$	$- 12$	$\frac{66}{5}$

Step 17.7

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

$(5) x - 12 + \frac{\frac{66}{5}}{x + \frac{3}{5}}$

Step 17.8

Simplify the quotient polynomial.

$5 x - 12 + \frac{66}{5 x + 3}$

Step 18

Since $- \frac{3}{5} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{3}{5}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{3}{5}$

Step 19

Apply synthetic division on $\frac{5 x^{2} - 9 x + 6}{x - 6}$ when $x = 6$ .

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Step 19.1

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

$6$	$5$	$- 9$	$6$

Step 19.2

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$6$	$5$	$- 9$	$6$

	$5$

Step 19.3

Multiply the newest entry in the result $(5)$ by the divisor $(6)$ and place the result of $(30)$ under the next term in the dividend $(- 9)$ .

$6$	$5$	$- 9$	$6$
		$30$
	$5$

Step 19.4

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

$6$	$5$	$- 9$	$6$
		$30$
	$5$	$21$

Step 19.5

Multiply the newest entry in the result $(21)$ by the divisor $(6)$ and place the result of $(126)$ under the next term in the dividend $(6)$ .

$6$	$5$	$- 9$	$6$
		$30$	$126$
	$5$	$21$

Step 19.6

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

$6$	$5$	$- 9$	$6$
		$30$	$126$
	$5$	$21$	$132$

Step 19.7

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

$(5) x + 21 + \frac{132}{x - 6}$

Step 19.8

Simplify the quotient polynomial.

$5 x + 21 + \frac{132}{x - 6}$

Step 20

Since $6 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $6$ is an upper bound for the real roots of the function.

Upper Bound: $6$

Step 21

Apply synthetic division on $\frac{5 x^{2} - 9 x + 6}{x + 6}$ when $x = - 6$ .

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Step 21.1

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

$- 6$	$5$	$- 9$	$6$

Step 21.2

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$- 6$	$5$	$- 9$	$6$

	$5$

Step 21.3

Multiply the newest entry in the result $(5)$ by the divisor $(- 6)$ and place the result of $(- 30)$ under the next term in the dividend $(- 9)$ .

$- 6$	$5$	$- 9$	$6$
		$- 30$
	$5$

Step 21.4

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

$- 6$	$5$	$- 9$	$6$
		$- 30$
	$5$	$- 39$

Step 21.5

Multiply the newest entry in the result $(- 39)$ by the divisor $(- 6)$ and place the result of $(234)$ under the next term in the dividend $(6)$ .

$- 6$	$5$	$- 9$	$6$
		$- 30$	$234$
	$5$	$- 39$

Step 21.6

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

$- 6$	$5$	$- 9$	$6$
		$- 30$	$234$
	$5$	$- 39$	$240$

Step 21.7

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

$(5) x - 39 + \frac{240}{x + 6}$

Step 21.8

Simplify the quotient polynomial.

$5 x - 39 + \frac{240}{x + 6}$

Step 22

Since $- 6 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 6$ is a lower bound for the real roots of the function.

Lower Bound: $- 6$

Step 23

Apply synthetic division on $\frac{5 x^{2} - 9 x + 6}{x + \frac{6}{5}}$ when $x = - \frac{6}{5}$ .

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Step 23.1

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

$- \frac{6}{5}$	$5$	$- 9$	$6$

Step 23.2

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$- \frac{6}{5}$	$5$	$- 9$	$6$

	$5$

Step 23.3

Multiply the newest entry in the result $(5)$ by the divisor $(- \frac{6}{5})$ and place the result of $(- 6)$ under the next term in the dividend $(- 9)$ .

$- \frac{6}{5}$	$5$	$- 9$	$6$
		$- 6$
	$5$

Step 23.4

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

$- \frac{6}{5}$	$5$	$- 9$	$6$
		$- 6$
	$5$	$- 15$

Step 23.5

Multiply the newest entry in the result $(- 15)$ by the divisor $(- \frac{6}{5})$ and place the result of $(18)$ under the next term in the dividend $(6)$ .

$- \frac{6}{5}$	$5$	$- 9$	$6$
		$- 6$	$18$
	$5$	$- 15$

Step 23.6

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

$- \frac{6}{5}$	$5$	$- 9$	$6$
		$- 6$	$18$
	$5$	$- 15$	$24$

Step 23.7

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

$(5) x - 15 + \frac{24}{x + \frac{6}{5}}$

Step 23.8

Simplify the quotient polynomial.

$5 x - 15 + \frac{120}{5 x + 6}$

Step 24

Since $- \frac{6}{5} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{6}{5}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{6}{5}$

Step 25

Determine the upper and lower bounds.

Upper Bounds: $2, 3, 6$

Lower Bounds: $- 1, - \frac{1}{5}, - 2, - \frac{2}{5}, - 3, - \frac{3}{5}, - 6, - \frac{6}{5}$

Step 26