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

Determining the bounds of a polynomial's zeros is a crucial aspect of polynomial analysis. It allows us to narrow down the potential real roots, making it easier to find them or approximate their values. In this article, we will delve into the process of verifying if a given value is an upper bound for the zeros of a polynomial function. Specifically, we'll examine the function f(x) = 4x³ - 12x² - x + 15 and assess whether 4 is indeed an upper bound for its zeros. To achieve this, we will utilize the Upper Bound Theorem, a powerful tool in polynomial algebra. This theorem provides a systematic approach to testing potential upper bounds by employing synthetic division and analyzing the resulting quotient and remainder. Our exploration will involve a step-by-step application of synthetic division, followed by a careful interpretation of the results in the context of the Upper Bound Theorem. By the end of this analysis, we will definitively determine whether 4 serves as an upper bound for the real zeros of the given polynomial function. Understanding the Upper Bound Theorem and its application is essential for anyone working with polynomials, as it simplifies the process of finding roots and understanding the behavior of polynomial functions.

Understanding the Upper Bound Theorem

The Upper Bound Theorem is a fundamental concept in polynomial algebra that provides a method for identifying an upper limit for the real zeros of a polynomial function. This theorem is particularly useful when trying to find the roots of a polynomial, as it helps to narrow down the search range. In essence, the Upper Bound 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 (i.e., they are all non-negative), then c is an upper bound for the real zeros of f(x). This means that there are no real zeros of the polynomial greater than c. The theorem hinges on the idea that if all the coefficients in the quotient and the remainder are non-negative, then plugging in any value greater than c into the polynomial will result in a positive value, thus precluding the possibility of a zero. This is because each term in the polynomial, when evaluated, will contribute positively, ensuring the overall result remains positive. Conversely, if there are sign changes in the quotient and remainder, it indicates that values greater than c might potentially lead to a zero, and thus c cannot be considered an upper bound. The Upper Bound Theorem is a powerful tool because it transforms the problem of finding an upper bound into a mechanical process of performing synthetic division and observing the signs of the resulting numbers. This makes it a practical and efficient method for polynomial analysis.

Applying Synthetic Division

To determine if 4 is an upper bound for the zeros of the polynomial f(x) = 4x³ - 12x² - x + 15, we will employ synthetic division. Synthetic division is a streamlined method for dividing a polynomial by a linear expression of the form (x - c). It simplifies the long division process by focusing on the coefficients of the polynomial and the value of c. In our case, c is 4. The process begins by writing down the coefficients of the polynomial, ensuring that all powers of x are accounted for (using a 0 as a placeholder if a term is missing). For f(x) = 4x³ - 12x² - x + 15, the coefficients are 4, -12, -1, and 15. We then set up the synthetic division table, placing c (which is 4) to the left and the coefficients to the right. The first step in the division process is to bring down the leading coefficient (4 in this case) below the line. Next, we multiply this number by c (4 * 4 = 16) and write the result under the next coefficient (-12). We then add these two numbers (-12 + 16 = 4) and write the sum below the line. This process is repeated for the remaining coefficients: multiply the latest result by c (4 * 4 = 16), write it under the next coefficient (-1), add them (-1 + 16 = 15), and write the sum below the line. Finally, multiply the latest result by c (15 * 4 = 60), write it under the last coefficient (15), and add them (15 + 60 = 75). The numbers below the line, excluding the last one, represent the coefficients of the quotient polynomial, and the last number (75) is the remainder. By carefully executing synthetic division, we obtain the necessary information to apply the Upper Bound Theorem and determine if 4 is indeed an upper bound for the zeros of f(x).

Interpreting the Results

After performing synthetic division with 4 on the polynomial f(x) = 4x³ - 12x² - x + 15, we obtained the following results: the coefficients of the quotient are 4, 4, and 15, and the remainder is 75. To determine if 4 is an upper bound for the zeros of f(x), we must now interpret these results in the context of the Upper Bound Theorem. The theorem states that if all the numbers in the bottom row of the synthetic division (the coefficients of the quotient and the remainder) are non-negative, then the test value (4 in this case) is an upper bound. Examining our results, we see that the numbers 4, 4, 15, and 75 are all positive. Since there are no negative numbers or sign changes, the condition of the Upper Bound Theorem is satisfied. This definitively tells us that 4 is indeed an upper bound for the real zeros of the polynomial f(x) = 4x³ - 12x² - x + 15. In practical terms, this means that there are no real roots of the polynomial that are greater than 4. This information is valuable when attempting to find the roots of the polynomial, as it narrows down the range of values we need to consider. It also provides insight into the behavior of the polynomial function, indicating that as x increases beyond 4, the function will not cross the x-axis again. Therefore, based on the successful application and interpretation of the Upper Bound Theorem, we can confidently conclude that 4 serves as an upper bound for the real zeros of the given polynomial.

Conclusion: Is 4 an Upper Bound?

In conclusion, after a thorough application of synthetic division and a careful interpretation of the results using the Upper Bound Theorem, we can definitively state that 4 is indeed an upper bound for the zeros of the function f(x) = 4x³ - 12x² - x + 15. The synthetic division process yielded a quotient with coefficients 4, 4, and 15, and a remainder of 75. The critical observation here is that all these values are non-negative. According to the Upper Bound Theorem, this absence of sign changes confirms that no real zero of the polynomial exists above the value of 4. This finding is significant in the context of polynomial analysis, particularly when seeking to identify the real roots of the function. By establishing 4 as an upper bound, we effectively reduce the search space for potential roots, making the root-finding process more efficient and targeted. Furthermore, this result offers valuable insight into the function's behavior, indicating that the graph of f(x) will not intersect the x-axis for any x-value greater than 4. This comprehensive analysis underscores the power and utility of the Upper Bound Theorem in understanding and characterizing polynomial functions. The ability to determine bounds for polynomial zeros is a fundamental skill in algebra and calculus, providing a crucial stepping stone for more advanced mathematical explorations.

Therefore, the answer is A. True