Find all the zeros, real and nonreal, of the polynomial [tex]p(x)=x^2-64 \pi^2$[/tex].

A. [tex]x=-64 \pi, x=64 \pi$[/tex]
B. [tex]x=64 \pi$[/tex]
C. [tex]x=-64 \pi$[/tex]
D. [tex]x=-8 \pi, x=8 \pi$[/tex]
E. [tex]x=8 \pi$[/tex]

Set the polynomial equal to zero: x 2 − 64 π 2 = 0 .
Isolate x 2 : x 2 = 64 π 2 .
Take the square root of both sides: x = ± 64 π 2 ​ .
Simplify to find the zeros: x = ± 8 π . The answer is x = − 8 π , x = 8 π ​ .

Explanation

Problem Analysis We are given the polynomial p ( x ) = x 2 − 64 π 2 and asked to find its zeros, which can be real or nonreal. To find the zeros, we need to solve the equation p ( x ) = 0 for x .

Setting up the Equation We set p ( x ) = 0 , which gives us the equation x 2 − 64 π 2 = 0.

Isolating x 2 Adding 64 π 2 to both sides of the equation, we get x 2 = 64 π 2 .

Solving for x Taking the square root of both sides, we obtain x = ± 64 π 2 ​ = ± 8 π .

Finding the Zeros Thus, the zeros of the polynomial are x = − 8 π and x = 8 π .

Final Answer The zeros of the polynomial p ( x ) = x 2 − 64 π 2 are x = − 8 π and x = 8 π . Therefore, the correct answer is d.


Examples
Understanding polynomial zeros is crucial in many areas, such as physics and engineering. For example, when designing a bridge, engineers need to analyze the stability of the structure, which can involve finding the zeros of a polynomial that describes the bridge's behavior under different loads. Similarly, in signal processing, zeros of polynomials are used to design filters that remove unwanted frequencies from a signal. In finance, understanding the roots of characteristic equations helps in analyzing the stability of economic models.

Answered by GinnyAnswer | 2025-07-04