Precalculus Examples

$f (x) = 2 x^{2} - 7$

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.

$p = \pm 1, \pm 7$

$q = \pm 1, \pm 2$

Step 1.2

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

$\pm 1, \pm \frac{1}{2}, \pm 7, \pm \frac{7}{2}$

Step 2

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

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Step 2.1

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

$7$	$2$	$0$	$- 7$

Step 2.2

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

$7$	$2$	$0$	$- 7$

	$2$

Step 2.3

Multiply the newest entry in the result $(2)$ by the divisor $(7)$ and place the result of $(14)$ under the next term in the dividend $(0)$ .

$7$	$2$	$0$	$- 7$
		$14$
	$2$

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.

$7$	$2$	$0$	$- 7$
		$14$
	$2$	$14$

Step 2.5

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

$7$	$2$	$0$	$- 7$
		$14$	$98$
	$2$	$14$

Step 2.6

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

$7$	$2$	$0$	$- 7$
		$14$	$98$
	$2$	$14$	$91$

Step 2.7

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

$(2) x + 14 + \frac{91}{x - 7}$

Step 2.8

Simplify the quotient polynomial.

$2 x + 14 + \frac{91}{x - 7}$

Step 3

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 4

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

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Step 4.1

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

$- 7$	$2$	$0$	$- 7$

Step 4.2

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

$- 7$	$2$	$0$	$- 7$

	$2$

Step 4.3

Multiply the newest entry in the result $(2)$ by the divisor $(- 7)$ and place the result of $(- 14)$ under the next term in the dividend $(0)$ .

$- 7$	$2$	$0$	$- 7$
		$- 14$
	$2$

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.

$- 7$	$2$	$0$	$- 7$
		$- 14$
	$2$	$- 14$

Step 4.5

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

$- 7$	$2$	$0$	$- 7$
		$- 14$	$98$
	$2$	$- 14$

Step 4.6

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

$- 7$	$2$	$0$	$- 7$
		$- 14$	$98$
	$2$	$- 14$	$91$

Step 4.7

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

$(2) x - 14 + \frac{91}{x + 7}$

Step 4.8

Simplify the quotient polynomial.

$2 x - 14 + \frac{91}{x + 7}$

Step 5

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 6

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

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Step 6.1

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

$\frac{7}{2}$	$2$	$0$	$- 7$

Step 6.2

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

$\frac{7}{2}$	$2$	$0$	$- 7$

	$2$

Step 6.3

Multiply the newest entry in the result $(2)$ by the divisor $(\frac{7}{2})$ and place the result of $(7)$ under the next term in the dividend $(0)$ .

$\frac{7}{2}$	$2$	$0$	$- 7$
		$7$
	$2$

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.

$\frac{7}{2}$	$2$	$0$	$- 7$
		$7$
	$2$	$7$

Step 6.5

Multiply the newest entry in the result $(7)$ by the divisor $(\frac{7}{2})$ and place the result of $(\frac{49}{2})$ under the next term in the dividend $(- 7)$ .

$\frac{7}{2}$	$2$	$0$	$- 7$
		$7$	$\frac{49}{2}$
	$2$	$7$

Step 6.6

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

$\frac{7}{2}$	$2$	$0$	$- 7$
		$7$	$\frac{49}{2}$
	$2$	$7$	$\frac{35}{2}$

Step 6.7

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

$(2) x + 7 + \frac{\frac{35}{2}}{x - \frac{7}{2}}$

Step 6.8

Simplify the quotient polynomial.

$2 x + 7 + \frac{35}{2 x - 7}$

Step 7

Since $\frac{7}{2} > 0$ and all of the signs in the bottom row of the synthetic division are positive, $\frac{7}{2}$ is an upper bound for the real roots of the function.

Upper Bound: $\frac{7}{2}$

Step 8

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

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Step 8.1

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

$- \frac{7}{2}$	$2$	$0$	$- 7$

Step 8.2

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

$- \frac{7}{2}$	$2$	$0$	$- 7$

	$2$

Step 8.3

Multiply the newest entry in the result $(2)$ by the divisor $(- \frac{7}{2})$ and place the result of $(- 7)$ under the next term in the dividend $(0)$ .

$- \frac{7}{2}$	$2$	$0$	$- 7$
		$- 7$
	$2$

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.

$- \frac{7}{2}$	$2$	$0$	$- 7$
		$- 7$
	$2$	$- 7$

Step 8.5

Multiply the newest entry in the result $(- 7)$ by the divisor $(- \frac{7}{2})$ and place the result of $(\frac{49}{2})$ under the next term in the dividend $(- 7)$ .

$- \frac{7}{2}$	$2$	$0$	$- 7$
		$- 7$	$\frac{49}{2}$
	$2$	$- 7$

Step 8.6

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

$- \frac{7}{2}$	$2$	$0$	$- 7$
		$- 7$	$\frac{49}{2}$
	$2$	$- 7$	$\frac{35}{2}$

Step 8.7

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

$(2) x - 7 + \frac{\frac{35}{2}}{x + \frac{7}{2}}$

Step 8.8

Simplify the quotient polynomial.

$2 x - 7 + \frac{35}{2 x + 7}$

Step 9

Since $- \frac{7}{2} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{7}{2}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{7}{2}$

Step 10

Determine the upper and lower bounds.

Upper Bounds: $7, \frac{7}{2}$

Lower Bounds: $- 7, - \frac{7}{2}$

Step 11