Problem 15 - College Algebra

Question

For the following exercises, identify the function as a power function, a polynomial function, or neither. $$ x\left(4-x^{2}\right)(2 x+1) $$

Step-by-Step Solution

Answer

The expression is a polynomial function of degree 4.

Step 1: Expand the Expression

First, we need to expand the given expression to make it easier to classify. The given expression is \( x(4 - x^2)(2x + 1) \). We'll expand it step-by-step. Start with \( (4 - x^2)(2x + 1) \). Use the distributive property: \( 4(2x) + 4(1) - x^2(2x) - x^2(1) \). This simplifies to \( 8x + 4 - 2x^3 - x^2 \). Next, multiply each term by \( x \): \( x \cdot 8x + x \cdot 4 - x \cdot 2x^3 - x \cdot x^2 \), resulting in \( 8x^2 + 4x - 2x^4 - x^3 \). Thus, the expanded expression is \( -2x^4 - x^3 + 8x^2 + 4x \).

Step 2: Classify as Polynomial or Not

The expanded expression is \( -2x^4 - x^3 + 8x^2 + 4x \). Look at the terms: \(-2x^4, -x^3, 8x^2, \) and \(4x\). All terms have variables raised to whole number powers (4, 3, 2, and 1), and there is no division by a variable or variable in the denominator. This means the expression is a polynomial.

Step 3: Determine the Type of Polynomial

This polynomial has four terms: \( -2x^4, -x^3, 8x^2, \) and \(4x\). The term with the highest degree is \(-2x^4\), which makes it a 4th degree polynomial. Therefore, it can also be classified as a polynomial function specifically of degree 4.

Key Concepts

Degree of Polynomial

The degree of a polynomial is an important concept that helps identify the characteristics of the polynomial function. In a polynomial expression, the degree is determined by the term with the highest power of the variable. For instance, in the expanded expression \[-2x^4 - x^3 + 8x^2 + 4x\], the term \(-2x^4\) has the highest power, which is 4.
This means the polynomial has a degree of 4. Understanding the degree is crucial because it dictates many of the polynomial's properties, including the number of potential roots and the general shape of its graph.
A higher degree indicates more complexity in behavior and more possibilities for turning points.

Terms of Polynomial

A polynomial is comprised of terms, each of which has a constant coefficient and one or more variables raised to whole number powers. Identifying and understanding the terms within a polynomial is necessary to work with these expressions effectively.
In the expression \[-2x^4 - x^3 + 8x^2 + 4x\], we have the following terms:

• \(-2x^4\) – the leading term, which determines the degree and the leading coefficient.
• \(-x^3\) – contributes to the polynomial's shape and possible inflection points.
• \(8x^2\) – affects the behavior of the polynomial in the interval around its roots.
• \(4x\) – the lower degree term that contributes to the linear behavior of the polynomial.

Each term plays a role in the behavior and characteristics of the polynomial function as a whole.

Expression Expansion

Expression expansion is a technique used to simplify complex polynomial products into a standard form, making them easier to analyze and understand. For example, with the expression \(x(4 - x^2)(2x + 1)\), the goal of expansion is to distribute all terms, eliminating multiplication signs between expressions.
Using the distributive property, you first expand \((4 - x^2)(2x + 1)\), giving \(8x + 4 - 2x^3 - x^2\). Then, multiply \(x\) with each term in the resulting expression to obtain \(-2x^4 - x^3 + 8x^2 + 4x\).
By expanding the expression, the polynomial is easier to work with when performing operations like addition, subtraction, or finding roots, because each term is explicitly defined.