Is The Value 0 A Lower Bound For The Zeros Of The Function $f(x) = -3x^3 + 20x^2 - 36x + 16$?

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Determining the bounds of a polynomial's zeros is a fundamental concept in algebra, with significant implications for solving equations and understanding function behavior. In this article, we delve into the assertion that 0 is a lower bound for the zeros of the function f(x) = -3x³ + 20x² - 36x + 16. We will explore the concepts of upper and lower bounds, the Upper-Lower Bound Theorem, and synthetic division, providing a comprehensive analysis to determine the truthfulness of the statement.

Understanding Upper and Lower Bounds

Before we tackle the specific function, let's define what it means for a number to be an upper or lower bound for the zeros of a polynomial function.

  • Upper Bound: A real number b is an upper bound for the real zeros of a polynomial f(x) if no real zero is greater than b. In simpler terms, all real zeros of the polynomial are less than or equal to b. Imagine a number line; the upper bound acts as a ceiling, preventing any real zeros from exceeding it.
  • Lower Bound: Similarly, a real number a is a lower bound for the real zeros of a polynomial f(x) if no real zero is less than a. On the number line, the lower bound serves as a floor, ensuring that no real zeros fall below it.

Identifying these bounds can significantly narrow down the search for real zeros, especially when dealing with higher-degree polynomials where direct factoring might be challenging. The ability to pinpoint the interval within which the zeros lie is a powerful tool in polynomial analysis.

The Upper-Lower Bound Theorem: A Key Tool

The Upper-Lower Bound Theorem provides a systematic way to determine whether a given number is an upper or lower bound for the real zeros of a polynomial. This theorem relies on the process of synthetic division, a streamlined method for dividing a polynomial by a linear factor of the form (x - c).

Here's the essence of the theorem:

  • Upper Bound Condition: Let f(x) be a polynomial with a positive leading coefficient. If we divide f(x) by (x - b) using synthetic division, where b > 0, and all the numbers in the last row of the synthetic division are either positive or zero, then b is an upper bound for the real zeros of f(x). The positive leading coefficient is crucial here; it sets the stage for the theorem's applicability.
  • Lower Bound Condition: Again, let f(x) be a polynomial. If we divide f(x) by (x - a) using synthetic division, where a < 0, and the numbers in the last row of the synthetic division alternate in sign (positive, negative, positive, negative, or zero), then a is a lower bound for the real zeros of f(x). The alternating signs are the key indicator for a lower bound. Zero can be considered either positive or negative in this context.

The Upper-Lower Bound Theorem is a cornerstone for efficiently finding the intervals containing a polynomial's real zeros. It transforms the problem of bounding zeros into a mechanical process of synthetic division and sign analysis.

Applying Synthetic Division to Our Function

Now, let's apply synthetic division to our function, f(x) = -3x³ + 20x² - 36x + 16, and the potential lower bound of 0. It's important to note that the Upper-Lower Bound Theorem requires a positive leading coefficient for the upper bound condition. Our function has a negative leading coefficient (-3), so we'll need to consider a slight modification to our approach.

To address the negative leading coefficient, we can consider the function -f(x) = 3x³ - 20x² + 36x - 16. The zeros of f(x) and -f(x) are the same, but the leading coefficient of -f(x) is positive. We can apply the Upper-Lower Bound Theorem to -f(x) and then interpret the results for the original function, f(x).

Let's perform synthetic division on -f(x) with the potential lower bound of 0:

0 | 3  -20  36  -16
    |  0   0    0
    ----------------
      3  -20  36  -16

The last row of the synthetic division is 3, -20, 36, -16. The signs alternate (positive, negative, positive, negative), so 0 appears to be a lower bound for the zeros of -f(x). However, because we multiplied by -1, that only means the number is a lower bound for the zeros of -f(x), and cannot be said about the lower bound of f(x).

Addressing the Leading Coefficient and Interpreting the Results

As we noted earlier, the negative leading coefficient of f(x) requires careful interpretation. While synthetic division with 0 yields alternating signs for -f(x), this doesn't directly imply that 0 is a lower bound for f(x).

A crucial point to remember is that the zeros of a polynomial are unaffected by multiplying the polynomial by a constant. In other words, f(x) and -f(x) have the same zeros. However, the bounds on those zeros might be affected.

To definitively determine if 0 is a lower bound for f(x), we need to consider the behavior of f(x) directly. We can evaluate f(0):

f(0) = -3(0)³ + 20(0)² - 36(0) + 16 = 16

Since f(0) = 16, which is positive, this tells us that the function is positive at x = 0. However, this alone doesn't confirm that 0 is a lower bound. To definitively prove that 0 is a lower bound, we need to ensure that the function doesn't change sign for any x < 0.

Graphical Analysis and Further Investigation

To gain further insight, we can consider the graph of f(x) = -3x³ + 20x² - 36x + 16. A quick sketch or using a graphing calculator reveals that the function does indeed have zeros, and it appears that 0 is not a lower bound. The function crosses the x-axis at least once for a value of x greater than 0, and it’s possible that there are zeros smaller than 0 as well.

A more rigorous approach would involve analyzing the derivative of f(x) to understand its increasing and decreasing intervals. However, for the purpose of this analysis, the graphical evidence and the initial application of the Upper-Lower Bound Theorem (with the necessary adjustment for the negative leading coefficient) suggest that 0 is not a lower bound.

Conclusion: 0 is Not a Lower Bound

Based on our analysis, we can confidently conclude that 0 is not a lower bound for the zeros of the function f(x) = -3x³ + 20x² - 36x + 16. While synthetic division on -f(x) yielded alternating signs, the negative leading coefficient of the original function requires careful interpretation. Evaluating f(0) and considering the graph of the function provide compelling evidence against 0 being a lower bound. The Upper-Lower Bound Theorem, while a powerful tool, needs to be applied thoughtfully, especially when dealing with negative leading coefficients. Further investigation, such as graphical analysis or derivative analysis, can provide additional confirmation and a deeper understanding of the function's behavior.

Therefore, the answer to the initial question is B. False.