Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, identify the degree of the polynomial. \(x^{2}+4 x+4\)

Step-by-Step Solution

Answer

The degree of the polynomial is 2.

Step 1: Identify the Terms of the Polynomial

The given polynomial is \(x^2 + 4x + 4\). In this step, identify the terms of the polynomial. The terms are \(x^2\), \(4x\), and \(4\).

Step 2: Determine the Degree of Each Term

Examine each term to find its degree. The term \(x^2\) has a degree of 2 because the exponent on \(x\) is 2. The term \(4x\) has a degree of 1 since the exponent on \(x\) is 1. The constant term \(4\) has a degree of 0 because it can be considered as \(4x^0\).

Step 3: Identify the Highest Degree

The degree of a polynomial is determined by the term with the highest degree. From Step 2, the highest degree among the terms \(x^2\), \(4x\), and \(4\) is 2.

Step 4: Conclusion

Therefore, the degree of the polynomial \(x^2 + 4x + 4\) is 2, as the highest degree term is \(x^2\).

Key Concepts

Polynomial Terms

When examining a polynomial, the very first step is to break it down into its fundamental pieces, known as "terms." A polynomial is composed of multiple terms, which are essentially the building blocks.

• Each term in a polynomial can be a variable raised to a power, a number, or both.
• For example, in the polynomial \(x^2 + 4x + 4\), the terms are \(x^2\), \(4x\), and \(4\).

These terms are separated by plus or minus signs, helping us to identify them easily. Recognizing terms properly is crucial, as it lays the foundation for the next steps in analyzing the polynomial's characteristics, such as determining its degree.

Exponents and Degree

Once the terms of a polynomial are identified, the next step is to understand the role of exponents in determining the degree of each term. The degree of a polynomial term is determined by the exponent of the variable in that term.

• The degree of \(x^2\) is 2, as the exponent on \(x\) is 2.
• In the term \(4x\), the exponent on \(x\) is 1, so the degree is 1.
• For the constant term \(4\), it might seem that there is no variable; however, you can consider it as \(4x^0\). Hence, the degree is 0 since \(x^0\) is always 1.

The degree of a single term is an indicator of its complexity in the polynomial. The highest degree term in a polynomial gives the overall degree of the polynomial itself.

Constant Term Degree

Many students wonder how to think of constant terms, such as the number \(4\) in our example polynomial \(x^2 + 4x + 4\). Even though it doesn’t look like it, constant terms do have a degree.

Here’s a simple way to understand it:

• A constant term can always be thought of as multiplying by \(x^0\). In our example, \(4\) is the same as \(4\times x^0\).
• This is because \(x^0\) equals one, and anything multiplied by one is itself, so \(4 = 4\times x^0\).
• Thus, the degree of the constant term \(4\) is 0.

Understanding that constant terms have a degree of zero ensures you don't overlook any term when finding the degree of a polynomial. Always remember, the degree of a polynomial is identified by the term with the highest degree.