Rational Root Theorem: Find The Function For -7/8

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Hey guys! Let's dive into the fascinating world of polynomials and the Rational Root Theorem. This theorem is a super handy tool for finding potential rational roots (or zeros) of polynomial functions. In this article, we're tackling the question: Which function has -7/8 as a potential rational root according to the Rational Root Theorem? We'll break down the theorem, apply it step-by-step, and figure out the right answer together. So, buckle up and let's get started!

Understanding the Rational Root Theorem

First things first, what exactly is the Rational Root Theorem? The Rational Root Theorem is a powerful tool that helps us identify potential rational roots of a polynomial equation. In simpler terms, it gives us a list of possible fractions that could be solutions to the equation. Remember, a rational root is simply a root that can be expressed as a fraction (p/q), where p and q are integers.

Here's the gist of the theorem: If a polynomial has integer coefficients, then every rational root of the polynomial can be written in the form p/q, where:

  • p is a factor of the constant term (the term without any variable).
  • q is a factor of the leading coefficient (the coefficient of the term with the highest power of the variable).

To really nail this down, let's break it down with an example. Suppose we have a polynomial like f(x) = ax^n + ... + c. According to the Rational Root Theorem, any potential rational root can be found by considering the factors of 'c' (the constant term) and the factors of 'a' (the leading coefficient). We then form all possible fractions by dividing the factors of 'c' by the factors of 'a'. These fractions are our potential rational roots. It's like casting a wide net to see which values might actually work when plugged back into the polynomial. This theorem doesn't guarantee that these potential roots are actual roots, but it significantly narrows down our search, making the process of finding roots much more manageable. Without it, we'd be guessing blindly, which can be a real headache, especially with higher-degree polynomials. So, by understanding and applying this theorem, we’re essentially setting ourselves up for success in solving polynomial equations.

Applying the Rational Root Theorem to -7/8

Now that we've got a handle on the theorem, let's apply it to our specific question: Which function has -7/8 as a potential rational root? Remember, -7/8 is in the form p/q, where p = -7 and q = 8. So, we need to find a function where 7 is a factor of the constant term and 8 is a factor of the leading coefficient.

Here's the breakdown:

  • -7 (or 7) must be a factor of the constant term.
  • 8 must be a factor of the leading coefficient.

Let's consider the options provided. We need to look for a function where the constant term has 7 as a factor and the leading coefficient has 8 as a factor. For example, if we have a polynomial f(x) = ax^n + ... + c, we're looking for 'c' that's divisible by 7 and 'a' that's divisible by 8. This is the core of our search strategy, and it aligns perfectly with the Rational Root Theorem's guidelines. By focusing on these two key parts of the polynomial, we can efficiently narrow down our options and identify the correct function. It's like being a detective and using clues to solve a mystery; in this case, the clues are the factors of -7/8, and the mystery is finding the right polynomial. So, let's get our detective hats on and start examining the candidates!

Analyzing the Given Functions

Let's say we have two functions to consider:

\bigcirc$ $f(x)=24 x^7+3 x^6+4 x^3-x-28

\bigcirc$ $f(x)=28 x^7+3 x^6+4x^3-x-24

Let's analyze the first function: f(x) = 24x^7 + 3x^6 + 4x^3 - x - 28. Here, the leading coefficient is 24, and the constant term is -28. Now, we need to check if 8 is a factor of 24 and if 7 is a factor of 28. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Aha! 8 is indeed a factor of 24. Next, the factors of 28 are 1, 2, 4, 7, 14, and 28. Bingo! 7 is a factor of 28. So, this function fits the criteria of the Rational Root Theorem with -7/8 as a potential rational root.

Now, let's move on to the second function: f(x) = 28x^7 + 3x^6 + 4x^3 - x - 24. In this case, the leading coefficient is 28, and the constant term is -24. We need to determine if 8 is a factor of 28 and if 7 is a factor of 24. The factors of 28 are 1, 2, 4, 7, 14, and 28. So, 8 is not a factor of 28. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Here, 7 is not a factor of 24. Therefore, this function does not meet the requirements for -7/8 to be a potential rational root according to the Rational Root Theorem. We've systematically broken down each function, checking the leading coefficient and constant term against the criteria set by our target rational root. This step-by-step approach allows us to confidently identify which function aligns with the theorem.

Identifying the Correct Function

Based on our analysis, we found that for f(x) = 24x^7 + 3x^6 + 4x^3 - x - 28, 8 is a factor of the leading coefficient (24), and 7 is a factor of the constant term (-28). This perfectly aligns with the Rational Root Theorem, confirming that -7/8 is indeed a potential rational root for this function.

On the other hand, for f(x) = 28x^7 + 3x^6 + 4x^3 - x - 24, we observed that 8 is not a factor of the leading coefficient (28), and 7 is not a factor of the constant term (-24). Therefore, according to the Rational Root Theorem, -7/8 cannot be a potential rational root for this function. Our methodical examination of both functions allows us to confidently zero in on the one that satisfies the conditions of the theorem. We've not only answered the question but also demonstrated the practical application of the Rational Root Theorem. This process highlights the theorem's effectiveness as a tool for narrowing down potential solutions and solving polynomial equations more efficiently.

Conclusion

So, there you have it! According to the Rational Root Theorem, -7/8 is a potential rational root of the function f(x) = 24x^7 + 3x^6 + 4x^3 - x - 28. We successfully navigated through the steps of the Rational Root Theorem, identified the factors, and matched them to the correct function. I hope this explanation has made the theorem a bit clearer and shown you how useful it can be for tackling polynomial problems. Keep practicing, and you'll become a pro at finding those rational roots! You got this, guys! This journey through the Rational Root Theorem illustrates how a systematic approach can demystify complex mathematical concepts. By breaking down the theorem into manageable steps and applying it to specific examples, we've not only found the correct answer but also deepened our understanding of polynomial functions. Remember, math isn't about memorizing formulas; it's about understanding the logic behind them. So, keep exploring, keep questioning, and most importantly, keep having fun with math!