Problem 22 - Precalculus : Building Concepts and Connections

Question

Determine whether the function is a polynomial function. If so, find the degree. If not, state the reason. $$g(x)=(x-1)^{2}$$

Step-by-Step Solution

Answer

Yes, the function \(g(x)=(x-1)^{2}\) is a polynomial function and its degree is 2.

Step 1: Identify whether the function is a polynomial

Start by examining the function \(g(x)=(x-1)^{2}\). Expanding this equation gives \(x^2 - 2x + 1\), which only involves addition and multiplication operations, and all exponents are non-negative integers. Thus, \(g(x)\) indeed meets the definition of a polynomial function.

Step 2: Determine the degree of the polynomial

After identifying that the function is a polynomial, the next step is to determine its degree. The degree of a polynomial is identified by the highest exponent in the equation. In the expanded function \(x^2 - 2x + 1\), the highest exponent is 2, hence the degree of the polynomial \(g(x)\) is 2.

Key Concepts

Degree of a Polynomial

Understanding the degree of a polynomial is crucial as it reveals essential characteristics about the behavior of the function, especially its end behavior and the maximum number of real zeros it can have. In simple terms, the degree of a polynomial is the highest power of the variable that appears in the function when it is expressed in its standard form. For instance, in the function

\( g(x)=(x-1)^{2} \),

when expanded, becomes \( x^2 - 2x + 1 \). Here, the term with the highest exponent is \( x^2 \), which tells us that the degree is 2. The degree informs us that the graph of this function will look like a parabola. It is also a cue that the polynomial function could potentially have up to two real zeroes. When reviewing problems of this nature, ensure that all exponents are non-negative integers to confirm that the function in question is indeed a polynomial.

Identifying Polynomials

Determining whether a given function is a polynomial involves a set of simple checks. A polynomial function is a mathematical expression consisting of variables and coefficients that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. For example, in our exercise

\( g(x)=(x-1)^{2} \),

we recognize it as a polynomial after expanding to \( x^2 - 2x + 1 \), as it fits the definition perfectly. To reiterate, key characteristics of polynomials include:

• One or more terms with variables raised to non-negative integer exponents
• The use of addition, subtraction, and multiplication (but not division by a variable)
• No variables in the denominator or under a root
• No complex mathematical operations like sine, cosine, or logarithms involved

An essential piece of advice when identifying polynomials is to look for these attributes in the standard form of the function, and if any of these criteria are not met, the function is not a polynomial.

Polynomial Equations

Polynomial equations are algebraic expressions that set a polynomial equal to a value, usually zero. Solving these equations requires finding the value(s) of the variable(s) that make the equation true. In our exercise, while \( g(x)=(x-1)^{2} \) is not presented as an equation to solve, if we were to set it equal to zero to find the roots, we would have a polynomial equation:

\[ (x-1)^{2} = 0 \]

To solve it, you would look for the values of \( x \) that satisfy the equation, which in this case are the points where the graph of the polynomial intersects the x-axis. The solutions to polynomial equations are fundamental in graphing the function and in understanding the function's behavior. It's worth noting that the Fundamental Theorem of Algebra assures us that a polynomial of degree \( n \) has exactly \( n \) complex roots (some of which may be real), counting multiplicities. Hence, with the degree of 2, as identified earlier, our \( g(x) \) will have two roots. Overall, being comfortable with solving polynomial equations is key to mastering the wider topic of polynomial functions.