Determining If 4 Is An Upper Bound For Zeros Of F(x)=4x³-12x²-x+15

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Is the value 4 an upper bound for the zeros of the function f(x) = 4x³ - 12x² - x + 15? This is a critical question in understanding the behavior and roots of polynomial functions. In this detailed exploration, we will delve into the concept of upper bounds for polynomial zeros, the method for determining them, and apply this knowledge to the given function. We will provide a step-by-step analysis to definitively answer whether 4 is indeed an upper bound for the zeros of the function. Let's embark on this mathematical journey to understand polynomial behavior and root determination.

Understanding Upper Bounds for Polynomial Zeros

To accurately address whether 4 is an upper bound for the zeros of the function f(x) = 4x³ - 12x² - x + 15, it is crucial to first grasp the concept of upper bounds in the context of polynomial functions. An upper bound for the real zeros of a polynomial function is a real number 'b' such that no real zero of the function is greater than 'b'. In simpler terms, if we find a number that acts as an upper boundary, then all real roots of the polynomial will be less than or equal to this number. Understanding this concept is the foundational step in our analysis. This helps us narrow down the possible range where real roots may exist, which is invaluable in various mathematical and practical applications. Identifying upper bounds is not just a theoretical exercise; it is a practical tool that streamlines the process of root finding and provides crucial insights into the function’s behavior. This concept is particularly useful in fields such as engineering, physics, and computer science, where polynomial functions are frequently used to model real-world phenomena. Therefore, understanding the upper bound allows for more efficient problem-solving and a deeper understanding of the systems being modeled. In the subsequent sections, we will delve into methods for determining these bounds, setting the stage for a focused analysis of the given function.

Methods to Determine Upper Bounds

To determine whether a given value is an upper bound for the real zeros of a polynomial function, we employ a method based on synthetic division. Synthetic division is an efficient algorithm for dividing a polynomial by a linear divisor of the form (x - c). The beauty of synthetic division lies in its ability to not only compute the quotient and remainder but also to provide critical information about the bounds of the polynomial's real roots. The process involves setting up a table with the coefficients of the polynomial and the potential upper bound value. We then perform the synthetic division, paying close attention to the resulting quotient and remainder. The key criterion for determining an upper bound is the sign of the numbers in the last row of the synthetic division table. If all the numbers in the last row (including the remainder) are either positive or zero, then the test value is an upper bound for the real zeros of the polynomial. This criterion is a direct consequence of the Remainder Theorem and the properties of polynomial division. It provides a straightforward and reliable method for identifying upper bounds. This method is not only efficient but also provides a clear visual indicator of whether a value qualifies as an upper bound. Mastering this technique is essential for anyone working with polynomial functions, as it simplifies the process of locating roots and understanding the function's behavior. In the next section, we will apply this method to the given function and test whether 4 is indeed an upper bound for its real zeros.

Applying Synthetic Division to f(x) = 4x³ - 12x² - x + 15 with a Test Value of 4

Now, let's apply the synthetic division method to the function f(x) = 4x³ - 12x² - x + 15 with the test value of 4. This will allow us to determine if 4 is an upper bound for the real zeros of the function. We begin by setting up the synthetic division table. Write down the coefficients of the polynomial: 4, -12, -1, and 15. Place the test value, 4, to the left. The synthetic division process unfolds as follows:

  1. Bring down the first coefficient (4) to the bottom row.
  2. Multiply the test value (4) by the number you just brought down (4), resulting in 16.
  3. Write this result (16) under the next coefficient (-12).
  4. Add -12 and 16, which gives 4. Write this sum in the bottom row.
  5. Multiply the test value (4) by the new number in the bottom row (4), resulting in 16.
  6. Write this result (16) under the next coefficient (-1).
  7. Add -1 and 16, which gives 15. Write this sum in the bottom row.
  8. Multiply the test value (4) by the new number in the bottom row (15), resulting in 60.
  9. Write this result (60) under the last coefficient (15).
  10. Add 15 and 60, which gives 75. Write this sum in the bottom row.

After completing the synthetic division, we examine the last row of the table. The numbers in the last row are 4, 4, 15, and 75. Notice that all these numbers are either positive or zero. This observation is crucial because, according to the upper bound criterion, if all the numbers in the last row are positive or zero, then the test value is an upper bound for the real zeros of the polynomial. Therefore, based on our synthetic division, we can conclude that 4 is indeed an upper bound for the real zeros of the function f(x) = 4x³ - 12x² - x + 15. This detailed application of the synthetic division method provides a clear and concise answer to the question at hand. In the following section, we will formally state our conclusion and reiterate the key steps that led us to it.

Conclusion: Is 4 an Upper Bound?

Having performed the synthetic division on the function f(x) = 4x³ - 12x² - x + 15 with the test value of 4, and observing that all the numbers in the last row of the synthetic division table are either positive or zero, we can definitively conclude that 4 is an upper bound for the real zeros of the function. This conclusion is a direct result of applying the upper bound theorem, which states that if synthetic division of a polynomial f(x) by (x - c), where c is a real number, results in all non-negative numbers in the quotient and remainder, then c is an upper bound for the real roots of f(x). Our analysis clearly demonstrates this principle in action. The synthetic division process not only provided us with the quotient and remainder but also gave us the critical information needed to determine the upper bound. This determination is significant because it narrows down the possible range of real roots for the polynomial. Knowing that 4 is an upper bound means that no real root of the function f(x) is greater than 4. This information is invaluable for further analysis, such as finding the actual roots or sketching the graph of the function. In practical applications, understanding the bounds of polynomial roots is essential for solving equations, optimizing functions, and modeling real-world phenomena. Our comprehensive analysis, from understanding the concept of upper bounds to applying synthetic division, has provided a clear and conclusive answer to the initial question. Therefore, we can confidently state that the value 4 is an upper bound for the zeros of the function f(x) = 4x³ - 12x² - x + 15.

Final Answer: A. True