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

Hey guys! Today, we're diving into a fun mathematical question: Is the value 4 an upper bound for the zeros of the function f(x) = 4x³ - 12x² - x + 15? This means we want to figure out if 4 is a number that's greater than or equal to all the real roots (or zeros) of this polynomial. Sounds interesting, right? Let's break it down and explore how we can solve this problem.

Understanding Upper Bounds and Zeros of Polynomials

Before we jump into the specifics of this function, let's quickly recap what we mean by "upper bound" and "zeros" in the context of polynomials. The zeros of a polynomial are the values of x that make the polynomial equal to zero. These are also known as roots or solutions of the polynomial equation f(x) = 0. Finding the zeros of a polynomial is a fundamental problem in algebra, and there are various techniques to do so, including factoring, the quadratic formula, and numerical methods.

Now, an upper bound for the zeros of a polynomial is a real number that is greater than or equal to the largest real zero of the polynomial. In other words, no real zero of the polynomial is larger than the upper bound. Similarly, a lower bound is a real number that is less than or equal to the smallest real zero. Identifying upper and lower bounds can help us narrow down the possible range of real roots, making it easier to find them. One common method for determining upper and lower bounds is the Upper and Lower Bound Theorem, which we'll be using shortly.

Applying the Upper and Lower Bound Theorem

The Upper and Lower Bound Theorem provides a systematic way to test potential upper and lower bounds for the real zeros of a polynomial. Here's how it works:

• Upper Bound Rule: Let f(x) be a polynomial with a positive leading coefficient. If we divide f(x) by (x - c), where c > 0, using synthetic division, and all the numbers in the last row (the quotient and the remainder) are either positive or zero, then c is an upper bound for the real zeros of f(x).
• Lower Bound Rule: Let f(x) be a polynomial. If we divide f(x) by (x - c), where c < 0, using synthetic division, and the numbers in the last row alternate in sign (with zero considered either positive or negative), then c is a lower bound for the real zeros of f(x).

To apply this theorem to our problem, we'll use synthetic division to divide the given polynomial, f(x) = 4x³ - 12x² - x + 15, by (x - 4), since we want to test if 4 is an upper bound. Synthetic division is a streamlined method for dividing a polynomial by a linear factor of the form (x - c). It involves using only the coefficients of the polynomial and the value of c to perform the division. It's a neat little trick that makes polynomial division much easier and faster!

Synthetic Division: The Key to Our Answer

Okay, let's get our hands dirty with some synthetic division! We're going to divide f(x) = 4x³ - 12x² - x + 15 by (x - 4). Here's how we set up the synthetic division:

4 | 4  -12  -1   15
  |____________________

First, we bring down the leading coefficient (4) to the bottom row:

4 | 4  -12  -1   15
  |____________________
    4

Next, we multiply the 4 (from the bottom row) by the test value 4 (the number to the left of the vertical line) and write the result (16) under the next coefficient (-12):

4 | 4  -12  -1   15
  |     16
  |____________________
    4

Now, we add -12 and 16, and write the sum (4) in the bottom row:

4 | 4  -12  -1   15
  |     16
  |____________________
    4   4

We continue this process: Multiply 4 (from the bottom row) by 4 (the test value) and write the result (16) under the next coefficient (-1):

4 | 4  -12  -1   15
  |     16  16
  |____________________
    4   4

Add -1 and 16, and write the sum (15) in the bottom row:

4 | 4  -12  -1   15
  |     16  16
  |____________________
    4   4   15

Finally, multiply 15 (from the bottom row) by 4 (the test value) and write the result (60) under the last coefficient (15):

4 | 4  -12  -1   15
  |     16  16  60
  |____________________
    4   4   15

Add 15 and 60, and write the sum (75) in the bottom row:

4 | 4  -12  -1   15
  |     16  16  60
  |____________________
    4   4   15  75

So, the last row of our synthetic division is 4, 4, 15, and 75. These numbers represent the coefficients of the quotient and the remainder when f(x) is divided by (x - 4).

Analyzing the Results

Now comes the crucial part: analyzing the results of our synthetic division. Remember, the Upper Bound Rule states that if all the numbers in the last row are either positive or zero, then the test value is an upper bound. Looking at our last row (4, 4, 15, 75), we see that all the numbers are indeed positive. Therefore, according to the Upper Bound Theorem, 4 is an upper bound for the real zeros of the function f(x) = 4x³ - 12x² - x + 15.

Conclusion: The Verdict is In!

So, what's the final answer? Based on our analysis using the Upper and Lower Bound Theorem and synthetic division, we can confidently say that 4 is indeed an upper bound for the zeros of the function f(x) = 4x³ - 12x² - x + 15. That means the statement in the original question is TRUE!

Isn't it cool how we can use these mathematical tools to figure out the behavior of polynomials? Synthetic division might seem a bit like a magic trick at first, but it's a powerful technique that can help us understand the roots and bounds of polynomial functions. Keep exploring, keep questioning, and keep having fun with math!