Problem 7 - College Algebra

Question

Determine which functions are polynomial functions. For those that are, identify the degree. $$ f(x)=x^{2}-3 x^{2}+5 $$

Step-by-Step Solution

Answer

The function \(f(x) = x^2 - 3x^2 + 5\) is a polynomial function and its degree is \(2\).

Step 1: Identify the structure of the function

The first step is to confirm whether the function \(f(x) = x^2 - 3x^2 + 5\) follows the general form of a polynomial function or not - if it can be expressed in the form \(a_nx^n + a_{n−1}x^{n−1} + \ldots + a_2x^2 + a_1x + a_0\), then it is a polynomial function. By looking at the function, it can be seen that it does follow this structure.

Step 2: Simplify the function

The function \(f(x) = x^2 - 3x^2 + 5\) can be simplified because the terms \(x^2\) and \(-3x^2\) can be combined into one term, which is \(-2x^2\). So, the function can be rewritten as \(f(x) = -2x^2 + 5\).

Step 3: Identify the degree of the function

The highest power of \(x\) in the simplified function \(f(x) = -2x^2 + 5\) is two (\(2\)). Hence, the degree of this polynomial function is \(2\).

Key Concepts

Polynomial Degree

Understanding the degree of a polynomial is pivotal when studying algebraic expressions. Consider the degree as the highest power of the variable within the polynomial after it has been properly simplified. To clarify, let's examine the polynomial function presented in the exercise:

\(f(x) = x^2 - 3x^2 + 5\). Our goal is to identify its degree. Initially, you might notice there are two terms involving the variable \(x\) raised to the second power. When we simplify, we combine like terms to consolidate our expression to \(f(x) = -2x^2 + 5\). Notice how we no longer have multiple \(x^2\) terms, but just one term involving \(x\). Now, since the highest power of \(x\) is 2, we can confidently say the degree of this polynomial is 2. This is significant because the degree of a polynomial can reveal a lot about its behavior, such as the number of roots it might have, and the general shape of its graph.

Simplifying Polynomials

Simplifying polynomials makes them easier to work with and understand. The process typically involves combining like terms and rearranging the expression into a standard form. A 'like term' refers to terms that have the same variable raised to the same power. In our example,

\(x^2\) and \(−3x^2\) are like terms because they share the same variable \(x\) to the same exponent, 2. To simplify, we sum the coefficients of these like terms: \(1x^2 + (-3)x^2\) simplifies to \(−2x^2\). Our expression then becomes \(f(x) = −2x^2 + 5\). This final simplified form is easier to evaluate, differentiate, or integrate. Simplification is not only about making the expression shorter; it often reveals more about the structure of the polynomial and its potential properties.

Algebraic Functions

An algebraic function is one that involves only algebraic operations, such as addition, subtraction, multiplication, division, and taking roots, on variables. Most school-level functions you'll encounter are algebraic and can often be represented by polynomials. The function from our exercise, \(f(x) = x^2 - 3x^2 + 5\), before simplification, and \(f(x) = -2x^2 + 5\) after, are examples of algebraic functions.

Importantly, notice how all the operations are sums, differences, and multiplications of variables raised to whole number exponents. Algebraic functions can take on many forms, but they don't involve operations that aren't algebraic, such as taking the sine or logarithm of a variable. Grasping this notion helps students understand the kind of operations permissible within the field of algebra and identify algebraic functions easily.