Precalculus Examples

$y = 2 - x^{2}$

Step 1

Reorder the polynomial $2 - x^{2}$ .

Tap for more steps...

Step 1.1

Reorder the polynomial.

$- x^{2} + 2$

Step 1.2

Apply the distributive property.

$- x^{2} \cdot - 1 + 2 \cdot - 1$

Step 1.3

Multiply $- x^{2} \cdot - 1$ .

Tap for more steps...

Step 1.3.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} + 2 \cdot - 1$

Step 1.3.2

Multiply $x^{2}$ by $1$ .

$x^{2} + 2 \cdot - 1$

Step 1.4

Multiply $2$ by $- 1$ .

$x^{2} - 2$

Step 2

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

Tap for more steps...

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.

$p = \pm 1, \pm 2$

$q = \pm 1$

Step 2.2

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

$\pm 1, \pm 2$

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

	$- 1$

Step 3.3

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 $(0)$ .

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

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.

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

Step 3.5

Multiply the newest entry in the result $(2)$ by the divisor $(- 2)$ and place the result of $(- 4)$ under the next term in the dividend $(2)$ .

$- 2$	$- 1$	$0$	$2$
		$2$	$- 4$
	$- 1$	$2$

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.

$- 2$	$- 1$	$0$	$2$
		$2$	$- 4$
	$- 1$	$2$	$- 2$

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.

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

Step 3.8

Simplify the quotient polynomial.

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

Step 4

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 5

Determine the upper and lower bounds.

No Upper Bounds

Lower Bound: $- 2$

Step 6