Is 3 An Upper Bound For Zeros Of F(x) = -3x³ + 20x² - 36x + 16

Determining the bounds of a polynomial's zeros is a fundamental concept in algebra and calculus. It allows us to narrow down the possible range of real roots, making it easier to find the actual solutions. In this article, we will delve into the process of verifying whether a given value is an upper bound for the zeros of a polynomial function. Specifically, we will examine the function $f(x) = -3x^3 + 20x^2 - 36x + 16$ and assess if 3 is indeed an upper bound for its zeros. To achieve this, we will employ the Upper Bound Theorem, a powerful tool in polynomial analysis. This theorem provides a systematic way to test potential upper bounds by using synthetic division. Let's embark on this journey to understand the intricacies of polynomial bounds and their significance in mathematical problem-solving.

Understanding the Upper Bound Theorem

Before we dive into the specifics of our function, let's first understand the Upper Bound Theorem. This theorem states that if we divide a polynomial $f(x)$ by $(x - c)$, where $c$ is a positive number, using synthetic division, and the resulting quotient and remainder have no sign changes, then $c$ is an upper bound for the real zeros of $f(x)$. In simpler terms, if all the numbers in the last row of the synthetic division are either positive or zero, then the value $c$ is an upper bound. This theorem is crucial because it provides a concrete method for testing potential upper bounds without having to find the actual roots of the polynomial. It's a valuable tool for narrowing down the search for zeros, especially when dealing with higher-degree polynomials where direct root-finding can be challenging. The theorem's effectiveness lies in its ability to quickly eliminate potential bounds, saving time and effort in the overall analysis of the polynomial's behavior. By understanding and applying the Upper Bound Theorem, we can gain significant insights into the nature and location of a polynomial's real roots.

Applying Synthetic Division to f(x) with c = 3

Now, let's apply the Upper Bound Theorem to our function, $f(x) = -3x^3 + 20x^2 - 36x + 16$, with the potential upper bound $c = 3$. We will use synthetic division to divide $f(x)$ by $(x - 3)$. Synthetic division is a streamlined method for dividing a polynomial by a linear factor, and it's particularly useful for applying the Upper Bound Theorem. Here's how we set up the synthetic division:

3 | -3  20  -36  16
  |__________________

We start by bringing down the leading coefficient (-3):

3 | -3  20  -36  16
  | -3
  |__________________

Next, we multiply the value we brought down (-3) by our test value (3) and write the result under the next coefficient (20):

3 | -3  20  -36  16
  |    -9
  | -3________________

Now, we add the numbers in the second column (20 and -9):

3 | -3  20  -36  16
  |    -9
  | -3  11____________

We repeat the process: multiply the result (11) by the test value (3) and write it under the next coefficient (-36):

3 | -3  20  -36  16
  |    -9  33
  | -3  11____________

Add the numbers in the third column (-36 and 33):

3 | -3  20  -36  16
  |    -9  33
  | -3  11  -3________

Finally, multiply the result (-3) by the test value (3) and write it under the last coefficient (16):

3 | -3  20  -36  16
  |    -9  33  -9
  | -3  11  -3________

Add the numbers in the last column (16 and -9):

3 | -3  20  -36  16
  |    -9  33  -9
  | -3  11  -3   7
  |__________________

The last row of our synthetic division is -3, 11, -3, and 7. Now, we need to analyze the signs of these numbers to determine if 3 is an upper bound.

Analyzing the Signs and Applying the Upper Bound Theorem

After performing the synthetic division, we obtained the last row: -3, 11, -3, and 7. To apply the Upper Bound Theorem, we need to examine the signs of these numbers. The sequence of signs is negative, positive, negative, and positive. This sequence indicates that there are sign changes in the last row. According to the Upper Bound Theorem, if there are any sign changes in the last row of the synthetic division, the test value is not an upper bound for the real zeros of the polynomial. In our case, we have sign changes between -3 and 11, and again between 11 and -3. Therefore, we can conclude that 3 is not an upper bound for the zeros of the function $f(x) = -3x^3 + 20x^2 - 36x + 16$. This conclusion is crucial for understanding the behavior of the polynomial and for guiding further analysis of its roots. The presence of sign changes provides valuable information about the distribution of the polynomial's zeros and helps us refine our search for these critical values.

Conclusion: Determining the Validity of the Statement

In conclusion, after performing synthetic division and analyzing the signs in the resulting row, we have determined that 3 is not an upper bound for the zeros of the function $f(x) = -3x^3 + 20x^2 - 36x + 16$. The statement "The value 3 is an upper bound for the zeros of the function shown below" is therefore false. This analysis highlights the importance of the Upper Bound Theorem as a tool for understanding the behavior of polynomial functions. By applying synthetic division and carefully examining the signs of the resulting coefficients, we can efficiently determine whether a given value serves as an upper bound for the real roots. This knowledge is not only valuable for solving mathematical problems but also for gaining a deeper understanding of the fundamental properties of polynomials. The process we've undertaken demonstrates a systematic approach to analyzing polynomial bounds, a skill that is essential for students and professionals alike in the fields of mathematics, engineering, and computer science.