Precalculus Examples

$f (x) = 13 x^{2} - 2$

Step 1

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

Tap for more steps...

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 13$

Step 1.2

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

$\pm 1, \pm \frac{1}{13}, \pm 2, \pm \frac{2}{13}$

Step 2

Apply synthetic division on $\frac{13 x^{2} - 2}{x - 1}$ when $x = 1$ .

Tap for more steps...

Step 2.1

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

$1$	$13$	$0$	$- 2$

Step 2.2

The first number in the dividend $(13)$ is put into the first position of the result area (below the horizontal line).

$1$	$13$	$0$	$- 2$

	$13$

Step 2.3

Multiply the newest entry in the result $(13)$ by the divisor $(1)$ and place the result of $(13)$ under the next term in the dividend $(0)$ .

$1$	$13$	$0$	$- 2$
		$13$
	$13$

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.

$1$	$13$	$0$	$- 2$
		$13$
	$13$	$13$

Step 2.5

Multiply the newest entry in the result $(13)$ by the divisor $(1)$ and place the result of $(13)$ under the next term in the dividend $(- 2)$ .

$1$	$13$	$0$	$- 2$
		$13$	$13$
	$13$	$13$

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.

$1$	$13$	$0$	$- 2$
		$13$	$13$
	$13$	$13$	$11$

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.

$(13) x + 13 + \frac{11}{x - 1}$

Step 2.8

Simplify the quotient polynomial.

$13 x + 13 + \frac{11}{x - 1}$

Step 3

Since $1 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $1$ is an upper bound for the real roots of the function.

Upper Bound: $1$

Step 4

Apply synthetic division on $\frac{13 x^{2} - 2}{x + 1}$ when $x = - 1$ .

Tap for more steps...

Step 4.1

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

$- 1$	$13$	$0$	$- 2$

Step 4.2

The first number in the dividend $(13)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$13$	$0$	$- 2$

	$13$

Step 4.3

Multiply the newest entry in the result $(13)$ by the divisor $(- 1)$ and place the result of $(- 13)$ under the next term in the dividend $(0)$ .

$- 1$	$13$	$0$	$- 2$
		$- 13$
	$13$

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.

$- 1$	$13$	$0$	$- 2$
		$- 13$
	$13$	$- 13$

Step 4.5

Multiply the newest entry in the result $(- 13)$ by the divisor $(- 1)$ and place the result of $(13)$ under the next term in the dividend $(- 2)$ .

$- 1$	$13$	$0$	$- 2$
		$- 13$	$13$
	$13$	$- 13$

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.

$- 1$	$13$	$0$	$- 2$
		$- 13$	$13$
	$13$	$- 13$	$11$

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.

$(13) x - 13 + \frac{11}{x + 1}$

Step 4.8

Simplify the quotient polynomial.

$13 x - 13 + \frac{11}{x + 1}$

Step 5

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 6

Apply synthetic division on $\frac{13 x^{2} - 2}{x - 2}$ when $x = 2$ .

Tap for more steps...

Step 6.1

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

$2$	$13$	$0$	$- 2$

Step 6.2

The first number in the dividend $(13)$ is put into the first position of the result area (below the horizontal line).

$2$	$13$	$0$	$- 2$

	$13$

Step 6.3

Multiply the newest entry in the result $(13)$ by the divisor $(2)$ and place the result of $(26)$ under the next term in the dividend $(0)$ .

$2$	$13$	$0$	$- 2$
		$26$
	$13$

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.

$2$	$13$	$0$	$- 2$
		$26$
	$13$	$26$

Step 6.5

Multiply the newest entry in the result $(26)$ by the divisor $(2)$ and place the result of $(52)$ under the next term in the dividend $(- 2)$ .

$2$	$13$	$0$	$- 2$
		$26$	$52$
	$13$	$26$

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.

$2$	$13$	$0$	$- 2$
		$26$	$52$
	$13$	$26$	$50$

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.

$(13) x + 26 + \frac{50}{x - 2}$

Step 6.8

Simplify the quotient polynomial.

$13 x + 26 + \frac{50}{x - 2}$

Step 7

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 8

Apply synthetic division on $\frac{13 x^{2} - 2}{x + 2}$ when $x = - 2$ .

Tap for more steps...

Step 8.1

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

$- 2$	$13$	$0$	$- 2$

Step 8.2

The first number in the dividend $(13)$ is put into the first position of the result area (below the horizontal line).

$- 2$	$13$	$0$	$- 2$

	$13$

Step 8.3

Multiply the newest entry in the result $(13)$ by the divisor $(- 2)$ and place the result of $(- 26)$ under the next term in the dividend $(0)$ .

$- 2$	$13$	$0$	$- 2$
		$- 26$
	$13$

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.

$- 2$	$13$	$0$	$- 2$
		$- 26$
	$13$	$- 26$

Step 8.5

Multiply the newest entry in the result $(- 26)$ by the divisor $(- 2)$ and place the result of $(52)$ under the next term in the dividend $(- 2)$ .

$- 2$	$13$	$0$	$- 2$
		$- 26$	$52$
	$13$	$- 26$

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.

$- 2$	$13$	$0$	$- 2$
		$- 26$	$52$
	$13$	$- 26$	$50$

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.

$(13) x - 26 + \frac{50}{x + 2}$

Step 8.8

Simplify the quotient polynomial.

$13 x - 26 + \frac{50}{x + 2}$

Step 9

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 10

Determine the upper and lower bounds.

Upper Bounds: $1, 2$

Lower Bounds: $- 1, - 2$

Step 11

Enter YOUR Problem