Problem 82 - Algebra and Trigonometry

Question

What are the zeros of a polynomial function and how are they found?

Step-by-Step Solution

Answer

The zeros of a polynomial function are the x-values that make the polynomial equal to zero. To find the zeros, we usually set the polynomial equal to zero and solve for x, which often requires factoring. In the example given, the zeros of the polynomial \(f(x) = x^2 - 5x + 6\) are \(x = 2\) and \(x = 3\).

Step 1: Definition of Zeros of a Polynomial Function

The zeros of a polynomial function are the x-values that make the polynomial equal to zero. These values are also called roots. If we have a polynomial \(f(x)\), then \(x = a\) is a zero of the polynomial if \(f(a) = 0\).

Step 2: Factoring the Polynomial

To find the zeros, set the polynomial equal to zero and solve for x. This usually involves factoring. For example, if our polynomial is \(f(x) = x^2 - 5x + 6\), we set it equal to zero and factor: \(0 = x^2 - 5x + 6\) which after factoring becomes \(0 = (x - 2)(x - 3)\).

Step 3: Solving for X

Take each factor that includes a variable and set it equal to zero, then solve. Therefore, when \(x - 2 = 0\), \(x = 2\) and when \(x - 3 = 0\), \(x = 3\). Those are the zeros of the polynomial.