Problem 1 - Algebra 2

Question

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. \(5 x^{6}-8 x^{2}\)

Step-by-Step Solution

Answer

The degree is 6, and the leading coefficient is 5.

Step 1: Identify the Polynomial

First, look at the given expression \(5x^6 - 8x^2\). It is a polynomial in one variable, specifically in the variable \(x\), because all the expressions have the same variable raised to whole number powers.

Step 2: Determine the Degree

The degree of a polynomial is the highest power of the variable \(x\). In the expression \(5x^6 - 8x^2\), the highest power of \(x\) is 6. Therefore, the degree of the polynomial is 6.

Step 3: Find the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree. In the polynomial \(5x^6 - 8x^2\), the term with the highest degree is \(5x^6\), and its coefficient is 5. So, the leading coefficient is 5.

Key Concepts

Degree of a Polynomial

In a polynomial, the *degree* is all about the highest power of the variable you see. Imagine you have multiple terms, each with the variable raised to a power. Among all these terms, the one with the largest exponent decides the degree. Let's take our example polynomial, \(5x^6 - 8x^2\). Here, \(x^6\) features the highest power, which is 6. Hence, this polynomial is of degree 6. Understanding the degree gives us insight into the polynomial's behavior. Its degree tells us how many solutions the polynomial might have and what its graph looks like. In our case, the degree is 6, which is an even number. This can suggest that the graph might extend in the same direction at both ends.

Leading Coefficient

The *leading coefficient* is like the leading actor in a movie; it's the main term's partner. In a polynomial, find the term with the highest power. Here, that term is \(5x^6\). The coefficient is the number in front of \(x^6\). So for our expression, the leading coefficient is 5. Understanding the leading coefficient is crucial because:

• It affects the shape and the direction of the graph of the polynomial.
• If the leading coefficient is positive, the ends of the polynomial's graph will point upwards (in even degrees).
• If negative, just imagine the graph pointing downwards.

By recognizing 5 as the leading coefficient, you can predict a lot about how this specific polynomial behaves!

Single Variable Polynomial

A *single variable polynomial* is a polynomial that uses only one variable throughout its expression. This is quite straightforward compared to polynomials that might have multiple variables, like \(x\) and \(y\). In our case, the polynomial \(5x^6 - 8x^2\) only uses the *single variable* \(x\). Identifying a polynomial as single variable is important because:

• It simplifies the analysis and calculations, as you're focusing on the behavior of just one variable.
• You only look at terms with the powers of that single variable.
• Simplifying such polynomials is often easier because there's no interaction between different variables.

In summary, single variable polynomials, like the one provided, are simpler to work with, allowing us to focus on fundamental polynomial concepts without the complexity of multiple variables.