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Understanding the zeros of a polynomial is crucial when working with polynomial equations. A zero of a polynomial is a solution to the equation when the polynomial is set equal to zero. In simpler terms, it is a value for which the entire polynomial equals 0. For example, in the polynomial presented, the zeros are 1, -2, and 3.
To find these, we express the polynomial using its zeros. So, the polynomial can be written in factor form, such as \[ P(x) = a(x - 1)(x + 2)(x - 3) \], where \( a \) is a constant multiplier. Each factor corresponds to a zero of the polynomial, making it simple to understand how zeros translate into factored expressions.
In general, if you have a polynomial of degree \( n \), there will be \( n \) zeros, assuming all zeros are real and distinct. Remember that zeros can sometimes be complex too!

Coefficients are the numerical parts of the terms in a polynomial. For a term like \( 3x^2 \), the coefficient is 3. Coefficients are vital because they define the weight or strength of each term within the polynomial.
In the context of modifying a polynomial, adjusting coefficients can help satisfy specific conditions, such as needing a specific term's coefficient to be a set value. In our exercise, the coefficient of \( x^2 \) was initially -2. To modify it to 3, we strategically chose a constant \( a \) which multiplied the entire polynomial to meet this condition. The equation \( a imes (-2) = 3 \) gives \( a = -\frac{3}{2} \), adjusting all terms proportionally without altering the zeros. This manipulation showcases the importance of understanding coefficients' roles in shaping the polynomial.

Polynomial expansion transforms a factored product into a sum of terms. This process involves distributing the factors into a single polynomial expression. Let's take a closer look at how this works!
Initially, we have \((x - 1)(x + 2)(x - 3)\). We first expand \((x - 1)(x + 2)\) to get \( x^2 + x - 2\). Next, multiply this result by the remaining \( (x - 3) \) to achieve \( x^3 - 2x^2 - 5x + 6 \).
Every step in expansion requires distributing each term of one factor across all terms of the other factor(s). This can get a bit complex, but with practice, polynomial expansion becomes a straightforward task. It is essential for transforming polynomial expressions into a format suitable for further manipulation or solving. Practice expanding polynomials to build confidence and mastery over this technique.

The degree of a polynomial is a central concept that tells us the highest power of the variable present in the polynomial expression. For instance, a polynomial \( x^3 - 2x^2 - 5x + 6 \) is said to have a degree of 3 because \( x^3 \) is the highest power.
Recognizing the degree is important, as it tells us about the number of zeros the polynomial might have. Specifically, a polynomial of degree \( n \) potentially has \( n \) zeros. Additionally, the degree also implies the polynomial's basic shape when graphed.
In the exercise, forming a degree 3 polynomial meant ensuring the product of linear factors yields terms that capture powers up to the cube term \( x^3 \). This characteristic of polynomials is not just a label; it directs what kind of roots (real or complex) we can expect, and what kind of transformations might influence the polynomial.