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Cubic polynomials are polynomials where the highest degree of the variable is 3. In our expression, this is identified by the term \( b^3 \). The term "cubic" refers to the fact that the variable \( b \) is raised to the third power. It is similar to how a cube has three dimensions. Understanding what makes a polynomial cubic is important because the degree of a polynomial influences its behavior and the methods used to solve or simplify it. A cubic polynomial such as \( b^3 - b^2 - 9b + 1 \) consists of four terms, each a different power of \( b \), ranging from \( b^3 \) to the constant term. These polynomials can often be part of larger algebraic operations like addition or subtraction, hence their importance in polynomial arithmetic.

Simplifying expressions is a fundamental skill in algebra that helps to make expressions easier to work with. When you are given an expression such as \( (b^3-b^2-9b+1) + (b^3-b^2-9b+1) \), the goal is to combine like terms. Since these two polynomials are identical, you can group them together by adding the coefficients of like terms. For instance:

• Combine \( b^3 \) terms: \( b^3 + b^3 = 2b^3 \)
• Combine \( -b^2 \) terms: \( -b^2 - b^2 = -2b^2 \)
• Combine \( -9b \) terms: \( -9b - 9b = -18b \)
• Combine constants: \( 1 + 1 = 2 \)

When simplified, the expression becomes \( 2b^3 - 2b^2 - 18b + 2 \). The process is all about combining terms to make a single, simplified polynomial expression.

Each polynomial has a degree, which is determined by the highest power of the variable in the expression. The degree tells us the nature of the polynomial's graph and how it behaves as the variable increases or decreases. Higher degree polynomials, like cubic ones with a degree of 3, can have more complex shapes. In our example, each polynomial \( b^3 - b^2 - 9b + 1 \) is degree 3 because of the \( b^3 \) term. Even after simplification: \( 2b^3 - 2b^2 - 18b + 2 \), the polynomial degree remains 3 since the term \( 2b^3 \) still dictates the highest degree. Understanding the degree helps in prediction and visualization of polynomial functions and solving for their roots.