Problem 23 - College Algebra

Question

Determine the end behavior of the functions. $$f(x)=x^{2}\left(2 x^{3}-x+1\right)$$

Step-by-Step Solution

Answer

The function grows towards \( +\infty \) as \( x \to +\infty \) and \( -\infty \) as \( x \to -\infty \).

Step 1: Understanding the Problem

We are tasked with determining the end behavior of the function \( f(x) = x^2 (2x^3 - x + 1) \). End behavior relates to how the function behaves as \( x \) approaches positive or negative infinity. Specifically, we are interested in the leading term of \( f(x) \) which will dictate the end behavior.

Step 2: Expand the Function

Expand the expression \( f(x) = x^2 (2x^3 - x + 1) \) to identify the terms. Distributing \( x^2 \) gives: \[ f(x) = x^2 imes 2x^3 + x^2 imes (-x) + x^2 imes 1 = 2x^5 - x^3 + x^2. \]

Step 3: Identify the Leading Term

From the expanded form \( f(x) = 2x^5 - x^3 + x^2 \), the leading term is \( 2x^5 \). The leading term is the term with the highest power of \( x \), which in this case is \( 2x^5 \). This leading term determines the end behavior of the polynomial.

Step 4: Analyze the Leading Term

The leading term is \( 2x^5 \). Since the power of \( x \) is odd (5) and the coefficient is positive (2), as \( x \to +fty \), \( f(x) \to +fty \), and as \( x \to -fty \), \( f(x) \to -fty \). This is consistent with the rules of polynomial behavior.

Key Concepts

Leading Term

The leading term is an essential component of understanding the behavior of polynomial functions, especially for determining their end behavior. In a polynomial expression, the leading term is the term with the highest power of the variable, which means it has the greatest degree. For example, in the expanded polynomial function \( f(x) = 2x^5 - x^3 + x^2 \), the leading term is \( 2x^5 \) because it has the largest exponent, which is 5.

The leading term is crucial because as \( x \) approaches very large positive or negative values (i.e., as \( x \to +\infty \) or \( x \to -\infty \)), the other terms become negligible in comparison. Thus, the leading term determines the general shape and direction of the graph of the polynomial as \( x \) grows larger in absolute value.

• If the highest power is even and the coefficient is positive, the graph will rise in both directions.
• If the highest power is even and the coefficient is negative, the graph will fall in both directions.
• If the highest power is odd and the coefficient is positive, the graph falls to the left and rises to the right.
• If the highest power is odd and the coefficient is negative, the graph rises to the left and falls to the right.

The analysis of the leading term provides immediate insight into the function’s end behavior, which is why it is often the primary focus when examining polynomials.

Polynomial Functions

Polynomial functions are mathematical expressions that involve a sum of powers of a variable multiplied by coefficients. A polynomial function can be written in the general form:

\( f(x) = a_nx^n + a_{n-1}x^{n-1} + \, \ldots \, + a_1x + a_0 \),
where \( a_n, a_{n-1}, \, \ldots \, , a_1, a_0 \) are constants, and \( n \) is a non-negative integer representing the highest power of \( x \).

These functions are fundamental in algebra because they encompass a wide range of real-world scenarios, from simple linear relationships to complex curves. Key characteristics of polynomial functions include:

• They have smooth, continuous graphs without breaks or holes.
• The degree of the polynomial (the highest power of the variable) dictates the number of turning points.
• The degree also determines the maximum number of roots or x-intercepts the polynomial can have.

Understanding polynomial functions allows us to model different phenomena and predict behaviors by analyzing roots, intercepts, and end behavior, which is crucial for calculus and various branches of science.

Expansion of Expressions

The expansion of expressions refers to multiplying out a factored expression to express it as a sum of terms. In the context of polynomial functions, expansion helps to identify and work with the individual terms of the polynomial. Take the function \( f(x) = x^2 (2x^3 - x + 1) \) as an example.

To expand this expression, we apply the distributive property, which involves multiplying each term inside the parentheses by \( x^2 \). Carrying out this expansion gives:
\[ f(x) = x^2 \times 2x^3 + x^2 \times (-x) + x^2 \times 1 = 2x^5 - x^3 + x^2. \]

Through this process, we transition from a compact, factored form to a detailed polynomial form. Expansion is particularly important for:

• Simplifying complex expressions, making them easier to handle.
• Identifying the leading term, which aids in analyzing end behavior.
• Facilitating further operations, such as differentiation or integration in calculus.

This skill is useful in various areas of mathematics and allows for a deeper understanding and manipulation of polynomial functions.