Precalculus Examples Polynomial and Rational Functions

Determine if Prime

$x^{2} - 49$

Step 1

Find the discriminant for $x^{2} - 49 = 0$ . In this case, $b^{2} - 4 a c = 196$ .

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Step 1.1

The discriminant of a quadratic is the expression inside the radical of the quadratic formula.

$b^{2} - 4 (a c)$

Step 1.2

Substitute in the values of $a$ , $b$ , and $c$ .

$0^{2} - 4 (1 \cdot - 49)$

Step 1.3

Evaluate the result to find the discriminant.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Raising $0$ to any positive power yields $0$ .

$0 - 4 (1 \cdot - 49)$

Step 1.3.1.2

Multiply $- 4 (1 \cdot - 49)$ .

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Step 1.3.1.2.1

Multiply $- 49$ by $1$ .

$0 - 4 \cdot - 49$

Step 1.3.1.2.2

Multiply $- 4$ by $- 49$ .

$0 + 196$

Step 1.3.2

Add $0$ and $196$ .

$196$

Step 2

A perfect square number is an integer that is the square of another integer. $\sqrt{196} = 14$ , which is an integer number.

$\sqrt{196} = 14$

Step 3

Since $196$ is the square of $14$ , it is a perfect square number.

$196$ is a perfect square number

Step 4

The polynomial $x^{2} - 49$ is not prime because the discriminant is a perfect square number.

Not prime

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Determine if Prime

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