Question

Find the degree and leading coefficient of the polynomial.\(2 x^{2}-x+1\)

Short answer

The degree of the polynomial is \(2\) and the leading coefficient is \(2\).

Step-by-step solution

Identify the degree of the polynomial

The term with the highest power of \(x\) in the given polynomial is \(2x^2\). The power attached to \(x\) in this term is \(2\), so the degree of the polynomial is \(2\).

Identify the leading coefficient of the polynomial

The coefficient of the term with the highest power of \(x\) (identified in Step 1) is considered the leading coefficient of the polynomial. For our polynomial \(2x^2-x+1\), the leading coefficient is \(2\).

Key concepts

Degree of a Polynomial

When we talk about the degree of a polynomial, we're referring to the highest power of the variable within the polynomial expression. Consider the polynomial given: \(2x^2 - x + 1\).
Here, the terms are \(2x^2\), \(-x\), and \(+1\). Each term includes a different power of \(x\), with \(2x^2\) having the highest power, which is \(2\).
Thus, the degree of this polynomial is \(2\).
Identifying the degree is crucial because it gives us insight into the polynomial's behavior and graph shape. For example:

• A polynomial of degree 2 is called a quadratic polynomial. This type will always graph as a parabola.
• The degree can help predict how many roots the polynomial might have, including complex roots.

Understanding the degree can help students comprehend more complex algebraic concepts and analyze polynomial functions better.

Leading Coefficient

The leading coefficient is equally important as it is the coefficient of the term with the highest degree in a polynomial. In our case, the polynomial function is \(2x^2 - x + 1\).
Here, the highest degree term is \(2x^2\), and the coefficient of this term is \(2\). This makes \(2\) the leading coefficient.
The leading coefficient provides valuable information, notably how the polynomial's graph behaves at the ends. Consider these points:

• If the leading coefficient is positive, like in our example, the ends of the polynomial's graph will rise.
• A negative leading coefficient often indicates the ends of the graph will fall.

These details about the leading coefficient are significant for graphing polynomials and predicting their long-term behavior.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. In the case of polynomials, algebra allows us to express equations like \(2x^2 - x + 1\) in a way that's solvable and analyzable.
There are several core elements in algebra that empower us to work with polynomial functions:

• Variables: Letters that represent unknown numbers or values. For example, \(x\) in \(2x^2 - x + 1\).
• Operations: Symbols representing mathematical processes; such as addition, subtraction, multiplication, and division.
• Coefficients: Numbers that multiply variables, like the \(2\) multiplying \(x^2\) in our polynomial.
• Constants: Fixed values without variables, such as \(+1\) in the polynomial.

Using the rules of algebra, students can solve for unknowns, factor polynomials, and manipulate equations for better understanding and practical applications.