Problem 10 - College Algebra

Question

Identify the function as a power function, a polynomial function, or neither. $$f(x)=2 x(x+2)(x-1)^{2}$$

Step-by-Step Solution

Answer

The function is a polynomial function.

Step 1: Expand the Expression

First, let's expand the expression to identify the type of function. \( f(x) = 2x(x+2)(x-1)^2 \). Start by expanding \((x+2)(x-1)^2\). The term \((x-1)^2\) is \(x^2 - 2x + 1\). Multiply this with \((x+2)\), resulting in \((x^2 - 2x + 1)(x+2) = x^3 + 2x^2 - 2x^2 - 4x + x + 2 = x^3 - x + 2\).

Step 2: Multiply by the Coefficient

Next, multiply the expanded expression by \(2x\). Each term in the polynomial \( (x^3 - x + 2) \) should be multiplied by \(2x\): \[2x \cdot (x^3 - x + 2) = 2x^4 - 2x^2 + 4x\]. Thus, the expression is \(f(x) = 2x^4 - 2x^2 + 4x\).

Step 3: Identify the Type of Function

Now that we have the expanded expression \(f(x) = 2x^4 - 2x^2 + 4x\), let's identify it. A polynomial function is defined as a sum of terms each of which is a product of a constant and a non-negative integer power of \(x\). Since the expression is such a sum, \(f(x)\) is a polynomial function.

Key Concepts

power functions

A power function is a type of mathematical function commonly described in the form \( f(x) = ax^n \), where \( a \) is a constant and \( n \) is a real number. The specific characteristic of a power function is that it consists of a single term with the variable base raised to a power.

**Key Characteristics of Power Functions:**

• The role of the exponent \( n \) is crucial. When \( n \) is a non-negative integer, the function can be part of a polynomial.
• If \( n \) equals 1, it represents a linear function. When \( n = 2 \), it becomes a quadratic function, and so forth for higher powers.
• For \( n \) being a negative number or a fraction, it is usually not classified as a polynomial.

For instance, \( f(x) = 2x^4 \) is a typical power function, with a single term where \( x \) is raised to a non-negative integer power.

In contrast, if you encounter a function with a variable exponent, like an exponential function, it would not fit as a power function. When analyzing expressions, identifying such a case helps determine the appropriate category in mathematical classifications.

function expansion

Function expansion is the process of taking a function expressed in a factored or simplified form and expanding it into a full polynomial form. This involves removing brackets and simplifying the terms to reveal its polynomial nature.

In our original exercise, we started with \( f(x) = 2x(x+2)(x-1)^2 \).
Through expansion, this becomes a polynomial by multiplying out all the factors.

**Steps in Function Expansion:**

• First, deal with any power or squared terms, such as expanding \( (x-1)^2 \) to \( x^2 - 2x + 1 \).
• Next, multiply the result with adjacent terms, like \((x+2)\), to further simplify the expression.
• Finally, multiply the entire expression by any remaining terms outside the brackets, in this case, \( 2x \).

Expansion provides a clearer insight into what type of function you are dealing with by fully exposing all terms and their respective degrees.

polynomial expression

A polynomial expression represents a sum of terms, each consisting of a constant multiplied by the variable raised to a non-negative integer power. These functions are key in algebra due to their simple yet flexible structure.

The polynomial \( f(x) = 2x^4 - 2x^2 + 4x \) fits this description perfectly:

• Each term, like \( 2x^4 \), features a constant coefficient (2) and a variable \( x \) raised to a non-negative integer power (4).
• The expression is considered polynomial since all exponents are integers and fall into non-negative categories.
• Polynomial expressions can universally accommodate real number solutions, given the degree of freedom offered by adding/subtracting terms of different degrees.

Additionally, the degree of a polynomial is the greatest exponent present across its terms. Here, the term \( 2x^4 \) determines that its degree is 4, indicating the polynomial is a quartic polynomial. To clearly define a function as polynomial, every term must meet the criterion of being multiplied by a variable raised to an integer power.

non-negative integer powers

In the context of polynomial functions, non-negative integer powers indicate that every term in the polynomial is a product of a constant and the variable raised only to whole numbers that are zero or positive. This concept is crucial for identifying both the degree and type of polynomial.

**Understanding Non-negative Integer Powers:**

• An integer power can take values such as 0, 1, 2, 3, and so on, meaning we never encounter terms like \( x^{-1} \) or \( x^{1.5} \) in a standard polynomial.
• For example, in \( f(x) = 2x^4 - 2x^2 + 4x \), the integer powers of \( x \) are 4, 2, and 1. All of these are non-negative integers.
• The absence of non-integer or negative exponents ensures that the polynomial's terms contribute smoothly to its overall behavior, without introducing undefined or asymptotic behavior seen in other function types.

Recognizing non-negative integer powers in functions helps not only in their classification but also in forming correct expectations of graphed characteristics, like continuity and smoothness over real numbers.