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The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. Essentially, these are the solutions to the polynomial equation. For example, if you have the polynomial function \( f(x) = x^2 - x - 2 \), the zeros are the points where \( f(x) = 0 \). In this solution, the zeros given were \(-1\) and \(2\). This means if you substitute \(x = -1\) or \(x = 2\) into the polynomial, the result will be zero: \[ f(-1) = (-1)^2 - (-1) - 2 = 0 \] and \[ f(2) = (2)^2 - 2 - 2 = 0 \]. To find the zeros of a polynomial, often you would set the polynomial equal to zero and solve for \(x\), which may involve factoring, using the quadratic formula, or other algebraic methods.

Factors of a polynomial are expressions that multiply together to give the polynomial. This is related to the concept of zeros because each zero corresponds to a factor. If \( c \) is a zero of a polynomial, then \( (x - c) \) is a factor of the polynomial.
If we have the zeros \(-1\) and \(2\), then the factors would be \( (x + 1) \) and \( (x - 2) \).
Why? Because \[ x + 1 = 0 \Rightarrow x = -1 \] and \[ x - 2 = 0 \Rightarrow x = 2. \]
To construct the polynomial from these factors, you multiply them together. So, from the given zeros, our polynomial factors are multiplied as follows:
\[ (x + 1)(x - 2). \]

Expanding polynomials involves multiplying the factors together to get the polynomial in its standard form. For the factors \( (x + 1) \) and \( (x - 2) \), we multiply them as follows:

\[ (x + 1)(x - 2) = x(x - 2) + 1(x - 2) \]
Distribute each term:
\[ = x^2 - 2x + x - 2 \]
Combine like terms:
\[ = x^2 - x - 2. \]
So, the polynomial in standard form is \( f(x) = x^2 - x - 2 \).
Understanding how to expand polynomials is crucial as it allows you to write the polynomial in its expanded form, making it easier to perform further operations like adding and subtracting polynomials, finding derivatives, and integrating polynomials.