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The roots of a polynomial are the values of the variable that make the polynomial equal to zero. For example, in the exercise, the given roots are –4 and 2. This means that when we substitute –4 or 2 into the polynomial, the result will be zero.

A polynomial with these roots will have factors that correspond to the reversed signs of these roots. Therefore, the polynomial will have factors \( (x + 4) \) and \( (x - 2) \).

Understanding the roots is crucial as it guides us in forming the polynomial equation by simply reversing their signs and including them as factors.

Let's take another example: If the roots were 3 and –5, the factors would be \( (x - 3) \) and \( (x + 5) \).

Factoring is the process of breaking down a polynomial into simpler 'factors' that, when multiplied together, give back the original polynomial.

In our exercise, we formed the factors from the given roots to get \( (x + 4) (x - 2) \).

This is the factored form of the polynomial.

Factoring is useful when solving polynomial equations or simplifying expressions. It’s like reverse-engineering; you start with the solutions (roots) and find the original polynomial.

Common methods of factoring include finding common factors, grouping, and using special formulas such as difference of squares. Practicing these methods helps in easily identifying and forming the factors.

To expand a polynomial means to multiply the factors to get a single polynomial expression.

In our example, we expanded \( (x + 4)(x - 2) \) by using the distributive property.

This property states that \( a(b + c) = ab + ac \).

So, we applied: \( (x + 4)(x - 2) = x(x - 2) + 4(x - 2) = x^2 - 2x + 4x - 8 \), which simplifies to: \[ x^2 + 2x - 8 \].

The expanded form is useful for identifying the degree and other characteristics of the polynomial.

It's also essential when adding or subtracting polynomials, as like terms must be combined. Expanding is the reverse process of factoring and helps in transitioning between different forms of polynomial expressions.