Polynomial Function Root Analysis Multiplicity And Leading Coefficient

Jul 23, 2025 by Admin 71 views

Decoding Polynomial Functions A Root of 0 with Multiplicity 1 and a Root of 2 with Multiplicity 4

Hey there, math enthusiasts! Let's dive into the fascinating world of polynomial functions. Today, we're going to break down a problem that involves roots, multiplicities, leading coefficients, and degrees. Trust me, it sounds complex, but we'll make it super clear and engaging. So, buckle up, and let's get started!

Understanding Polynomial Functions

Before we tackle the main problem, let's ensure we're all on the same page regarding the fundamentals of polynomial functions. A polynomial function is essentially an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. These functions can be written in the general form:

f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0

Where:

f(x) represents the polynomial function.
x is the variable.
a_n, a_{n-1}, ..., a_1, a_0 are the coefficients (real numbers).
n is a non-negative integer representing the degree of the polynomial.

The degree of the polynomial is the highest power of x (in this case, n). The leading coefficient is the coefficient of the term with the highest degree (here, a_n).

Roots and Multiplicity

Now, let's talk about roots. The roots of a polynomial function are the values of x for which f(x) = 0. In simpler terms, they are the points where the graph of the function intersects the x-axis. Each root has a property called multiplicity, which tells us how many times that root appears in the factored form of the polynomial.

For example, if a polynomial has a factor of (x - c)^m, then c is a root with multiplicity m. The multiplicity affects the behavior of the graph at the root. If the multiplicity is odd, the graph crosses the x-axis at that root. If the multiplicity is even, the graph touches the x-axis but doesn't cross it (it turns around).

Leading Coefficient and Degree

The leading coefficient and the degree of the polynomial play crucial roles in determining the end behavior of the function. The end behavior refers to what happens to the function as x approaches positive or negative infinity.

Leading Coefficient: If the leading coefficient is positive, the function will tend towards positive infinity as x goes to positive infinity. If it's negative, the function will tend towards negative infinity as x goes to positive infinity.
Degree: If the degree is even, both ends of the graph will go in the same direction (either both up or both down). If the degree is odd, the ends will go in opposite directions.

Understanding these fundamentals is crucial for tackling the problem at hand. We'll use these concepts to dissect the given polynomial function and determine its behavior.

Problem Breakdown: A Detailed Analysis

Okay, guys, let's get into the meat of the problem. We're given a polynomial function with some specific characteristics. The main goal here is to really take the time to thoroughly break down each piece of information provided. Let's methodically examine the core components of the problem so we're all on the same page.

Root of 0 with Multiplicity 1: This tells us that x = 0 is a root, and it appears once in the factored form of the polynomial. This means the polynomial has a factor of (x - 0)^1, which is simply x. Because the multiplicity is 1 (an odd number), the graph will cross the x-axis at x = 0.
Root of 2 with Multiplicity 4: Here, x = 2 is a root that appears four times in the factored form. This means the polynomial has a factor of (x - 2)^4. Since the multiplicity is 4 (an even number), the graph will touch the x-axis at x = 2 but not cross it.
Negative Leading Coefficient: This is a big clue about the end behavior. A negative leading coefficient means that as x goes to positive infinity, the function will go to negative infinity. In mathematical terms, as x → ∞, f(x) → -∞.
Odd Degree: An odd degree tells us that the ends of the graph will go in opposite directions. Since we already know the function goes to negative infinity as x goes to positive infinity, this means that as x goes to negative infinity, the function must go to positive infinity. So, as x → -∞, f(x) → ∞.

Constructing the Polynomial

Now, let's put these pieces together to construct a general form of the polynomial. We know it has factors of x and (x - 2)^4. So, the polynomial can be written in the form:

f(x) = a * x * (x - 2)^4

Where a is the leading coefficient. We know that a is negative because the problem states that the leading coefficient is negative. The degree of this polynomial is 1 + 4 = 5, which is odd, as specified.

Analyzing the Intervals

To determine where the function is positive or negative, we need to analyze the intervals created by the roots. The roots are x = 0 and x = 2. These roots divide the number line into three intervals:

(-∞, 0)
(0, 2)
(2, ∞)

We'll examine the sign of f(x) in each interval.

Determining the Sign of the Function in Each Interval

Alright, let's get our hands dirty and figure out the sign of our function in each interval. This is where we see how the roots and multiplicities really affect the behavior of the polynomial. By understanding whether the function is positive or negative in these intervals, we can get a clear picture of its overall graph and behavior.

Interval 1: (-∞, 0)

In this interval, x is negative. Let's pick a test value, say x = -1. Plugging this into our polynomial form:

f(x) = a * x * (x - 2)^4

We get:

f(-1) = a * (-1) * (-1 - 2)^4 = a * (-1) * (-3)^4 = a * (-1) * 81 = -81a

Since a is negative, -81a will be positive. Therefore, the function is positive in the interval (-∞, 0). This also means that as we approach 0 from the left, the function's graph is above the x-axis.

Interval 2: (0, 2)

Here, x is positive, but less than 2. Let's choose a test value, like x = 1. Substituting into our polynomial:

f(1) = a * (1) * (1 - 2)^4 = a * (1) * (-1)^4 = a * (1) * 1 = a

Since a is negative, f(1) is negative. Thus, the function is negative in the interval (0, 2). The graph will be below the x-axis in this interval.

Interval 3: (2, ∞)

In this interval, x is greater than 2. Let's pick x = 3 as our test value:

f(3) = a * (3) * (3 - 2)^4 = a * (3) * (1)^4 = 3a

Since a is negative, 3a is also negative. The function is negative in the interval (2, ∞). This aligns with our knowledge that the function tends towards negative infinity as x goes to positive infinity.

Summarizing the Signs

Let's put it all together:

(-∞, 0): f(x) is positive
(0, 2): f(x) is negative
(2, ∞): f(x) is negative

Now, with this sign analysis in hand, we can evaluate the given statements and determine which ones hold true. We have a solid understanding of how our polynomial function behaves across the number line.

Evaluating the Given Statements

Now that we've thoroughly analyzed our polynomial function, let's circle back to the original statements and see which ones are true. This is where all our hard work pays off, and we can confidently draw conclusions based on our understanding of the function's behavior.

Statement A: The function is positive on (-∞, 0).

Based on our interval analysis, we found that f(x) is indeed positive in the interval (-∞, 0). When we plugged in a test value of x = -1, we saw that f(-1) = -81a, and since a is negative, -81a is positive. Therefore, this statement is true.

To solidify this further, remember that as x approaches negative infinity, the function goes to positive infinity because the degree is odd and the leading coefficient is negative. This confirms that the function is positive as we move towards negative infinity.

Final Answer

So, after a comprehensive analysis, we've determined that:

Statement A: The function is positive on (-∞, 0) - TRUE

We systematically broke down the problem, understood the significance of each piece of information, and used that knowledge to analyze the behavior of the polynomial function. This approach not only helps us solve this specific problem but also equips us with the tools to tackle similar challenges in the future. Keep practicing, and you'll become a polynomial pro in no time!