Problem 49 - Algebra 2

Question

Explain why a constant polynomial such as \(f(x)=4\) has degree 0 and a linear polynomial such as \(f(x)=x+5\) has degree 1

Step-by-Step Solution

Answer

A constant polynomial has degree 0 as it's effectively \(x^0\); a linear polynomial has degree 1 with the term \(x^1\).

Step 1: Understanding Polynomial Degree

The degree of a polynomial is the highest power of the variable that appears in the polynomial with a non-zero coefficient. Essentially, it's about finding which term dictates the polynomial's degree.

Step 2: Analyzing the Constant Polynomial

For the constant polynomial, such as \(f(x) = 4\), there is actually an implied \(x^0\). This is because any number is \(x\) to the power of zero since \(x^0 = 1\). Since there is no higher power of \(x\) present, the degree is 0.

Step 3: Analyzing the Linear Polynomial

For the linear polynomial \(f(x) = x + 5\), we see two terms: \(x\) and a constant (5). The term with the highest power of \(x\) is the first one, \(x^1\). Therefore, the degree of this polynomial is 1 because \(x\) is equivalent to \(x^1\).

Key Concepts

Constant Polynomial

A constant polynomial is a type of polynomial where the value remains the same no matter what the variable value is. It is represented simply as a constant number — for example, 4, 7, or -3. In mathematical terms, such a polynomial can be written as \( f(x) = c \), where \( c \) is a constant, and without any variable attached to it by default.
Even if it seems that there are no variables present, mathematicians view it as having the term \( x^0 \) because any number to the power of zero is 1. This is why you might not see \( x \) explicitly, but it is understood to be there in the expression \( c \cdot x^0 \).

• For example, if we take \( f(x) = 4 \), this implies \( f(x) = 4 \times x^0 \), as \( x^0 = 1 \).
• Since there's no other term with a higher power of \( x \), the degree of a constant polynomial is zero.

This indicates the lack of variability with respect to \( x \), making it a consistent value across all inputs.

Linear Polynomial

A linear polynomial is a polynomial of the first degree, which means it has the highest variable power of 1. The typical format for a linear polynomial is \( f(x) = ax + b \), where \( a \) and \( b \) are constants.
A simple example is \( f(x) = x + 5 \). Here, the expression consists of two terms: \( x \) and a constant, 5. In this polynomial, the highest power of \( x \) is one. This means it is a first-degree polynomial or a linear polynomial.

• The term \( x \) can be seen as \( x^1 \), which emphasizes its degree. The constant 5 does not affect the degree because constants are thought of as variables raised to the power of zero.
• For a polynomial to be linear, it should have at most one variable which should not be raised to any power higher than 1.

This form establishes a straight-line graph when plotted, hence the name "linear." It represents a rate of change that remains constant, showcasing the relationship between the variable and its impact on the polynomial's value.

Highest Power of Variable

The highest power of a variable in a polynomial is the most critical factor in determining the polynomial's degree. The degree indicates the most significant rate of change within that polynomial.
In polynomial expressions, terms are composed of coefficients and variables. The degree is pinpointed by identifying the term with the largest exponent of the variable. Each aspect of the polynomial contributes to shaping the overall graph and behavior outlined by the expression.

• If the polynomial is \( f(x) = x^3 + 2x^2 + 5 \), the highest power of \( x \) is 3, therefore the degree is 3.
• In a constant polynomial like \( f(x) = 4 \), even though variables don't seem evident, the implicit power is zero.
• For a linear polynomial such as \( f(x) = x + 5 \), the highest power is 1.

Determining the degree by identifying the highest power in terms of \( x \) gives insight into the polynomial's shape, nature, and type. Hence, understanding this fundamental aspect is key in comprehending polynomial expressions.