Precalculus Examples

$f (x) = 2 x - 3$

Step 1

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

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Step 1.1

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.

$p = \pm 1, \pm 3$

$q = \pm 1, \pm 2$

Step 1.2

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

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

Step 2

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

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Step 2.1

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

$- 1$	$2$	$- 3$

Step 2.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$2$	$- 3$

	$2$

Step 2.3

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

$- 1$	$2$	$- 3$
		$- 2$
	$2$

Step 2.4

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

$- 1$	$2$	$- 3$
		$- 2$
	$2$	$- 5$

Step 2.5

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

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

Step 2.6

Simplify the quotient polynomial.

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

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 $\frac{2 x - 3}{x + \frac{1}{2}}$ when $x = - \frac{1}{2}$ .

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Step 4.1

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

$- \frac{1}{2}$	$2$	$- 3$

Step 4.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- \frac{1}{2}$	$2$	$- 3$

	$2$

Step 4.3

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

$- \frac{1}{2}$	$2$	$- 3$
		$- 1$
	$2$

Step 4.4

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

$- \frac{1}{2}$	$2$	$- 3$
		$- 1$
	$2$	$- 4$

Step 4.5

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

$2 + \frac{- 4}{x + \frac{1}{2}}$

Step 4.6

Simplify the quotient polynomial.

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

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 $\frac{2 x - 3}{x - 3}$ when $x = 3$ .

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Step 6.1

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

$3$	$2$	$- 3$

Step 6.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$3$	$2$	$- 3$

	$2$

Step 6.3

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

$3$	$2$	$- 3$
		$6$
	$2$

Step 6.4

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

$3$	$2$	$- 3$
		$6$
	$2$	$3$

Step 6.5

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

$2 + \frac{3}{x - 3}$

Step 7

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 8

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

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Step 8.1

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

$- 3$	$2$	$- 3$

Step 8.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- 3$	$2$	$- 3$

	$2$

Step 8.3

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

$- 3$	$2$	$- 3$
		$- 6$
	$2$

Step 8.4

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

$- 3$	$2$	$- 3$
		$- 6$
	$2$	$- 9$

Step 8.5

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

$2 + \frac{- 9}{x + 3}$

Step 8.6

Simplify the quotient polynomial.

$2 - \frac{9}{x + 3}$

Step 9

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 10

Apply synthetic division on $\frac{2 x - 3}{x - \frac{3}{2}}$ when $x = \frac{3}{2}$ .

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Step 10.1

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

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

Step 10.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

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

	$2$

Step 10.3

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

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

Step 10.4

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

$\frac{3}{2}$	$2$	$- 3$
		$3$
	$2$	$0$

Step 10.5

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

$2$

Step 11

Since $\frac{3}{2} > 0$ and all of the signs in the bottom row of the synthetic division are positive, $\frac{3}{2}$ is an upper bound for the real roots of the function.

Upper Bound: $\frac{3}{2}$

Step 12

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

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Step 12.1

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

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

Step 12.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

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

	$2$

Step 12.3

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

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

Step 12.4

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

$- \frac{3}{2}$	$2$	$- 3$
		$- 3$
	$2$	$- 6$

Step 12.5

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

$2 + \frac{- 6}{x + \frac{3}{2}}$

Step 12.6

Simplify the quotient polynomial.

$2 - \frac{12}{2 x + 3}$

Step 13

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 14

Determine the upper and lower bounds.

Upper Bounds: $3, \frac{3}{2}$

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

Step 15