2. Give an example of each of the following.
b. A repeated linear factor

Verified step by step guidance

1. Understand that a repeated linear factor in a polynomial is a linear factor (of the form \( (x - a) \)) that appears more than once, meaning it is raised to a power greater than 1.
2. Recall that a linear factor looks like \( (x - a) \), where \( a \) is a constant.
3. To create a repeated linear factor, take a linear factor and raise it to a power greater than 1, for example, \( (x - 3)^2 \).
4. An example polynomial with a repeated linear factor could be \( (x - 3)^2 (x + 1) \), where \( (x - 3) \) is the repeated linear factor.
5. This polynomial shows the repeated linear factor clearly, as \( (x - 3) \) appears twice, indicating multiplicity 2.

Key Concepts

Polynomial Factors

Polynomial factors are expressions that multiply together to form a polynomial. Understanding how to factor polynomials into linear and nonlinear components is essential for simplifying and solving polynomial equations.

Linear Factors

A linear factor is a polynomial of degree one, typically in the form (x - a), where 'a' is a root of the polynomial. Recognizing linear factors helps in breaking down polynomials into simpler parts.

Repeated Factors

A repeated factor occurs when a factor appears more than once in the factorization of a polynomial, such as (x - a)^n with n > 1. Identifying repeated factors is important for understanding multiplicity of roots and their effects on the graph.