Unlocking Polynomials: Zeros And Factoring Explained

Hey guys! Today, we're diving deep into the world of polynomials, specifically the function f(x) = 15x³ + 107x² - 247x + 45. Our mission, should we choose to accept it, is to (a) list all possible rational zeros, (b) find all the actual rational zeros, and (c) factor this bad boy completely. Buckle up, because we're about to embark on a mathematical adventure!

Listing Possible Rational Zeros: The Rational Root Theorem to the Rescue

So, how do we even begin to find the zeros of a polynomial? That's where the Rational Root Theorem comes in handy. This theorem is like a treasure map, guiding us to potential rational zeros. It states that if a polynomial has integer coefficients, then every rational zero of the polynomial has the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. Think of it as a systematic way to narrow down our search, making the whole process much less daunting. Understanding and applying the Rational Root Theorem is a fundamental skill in polynomial algebra, and it forms the backbone of many zero-finding techniques. It is a cornerstone concept that allows us to make educated guesses about the possible rational roots of a polynomial equation, thereby simplifying the process of finding actual solutions. Without this theorem, we would be left to essentially guess and check, a method that is both time-consuming and often unproductive.

In our case, the constant term is 45, and its factors (the p values) are ±1, ±3, ±5, ±9, ±15, and ±45. The leading coefficient is 15, and its factors (the q values) are ±1, ±3, ±5, and ±15. Now, we need to list all possible fractions p/q. This is where things can seem a little overwhelming, but stay with me! We systematically divide each p value by each q value, simplifying fractions as we go. Remember, we're looking for rational roots, which are numbers that can be expressed as a fraction of two integers. The Rational Root Theorem provides a comprehensive list of potential candidates, ensuring that we don't miss any possibilities. By methodically working through the factors of the constant term and the leading coefficient, we create a pool of potential solutions that we can then test using methods like synthetic division or direct substitution.

When we do this, we get a list of possible rational zeros: ±1, ±3, ±5, ±9, ±15, ±45, ±1/3, ±5/3, ±1/5, ±3/5, ±9/5, and ±45/5 (which simplifies to ±9, already in our list), ±15/3 (which simplifies to ±5, already in our list), ±45/15 (which simplifies to ±3, already in our list). Phew! That's a lot of possibilities, but don't worry; we won't have to test them all. The Rational Root Theorem gives us a starting point, a manageable set of potential solutions to explore. Without this theorem, we would be facing an infinite number of possibilities, making the task of finding zeros practically impossible. So, let's appreciate the power of this theorem and move on to the next step in our polynomial-solving adventure.

Finding the Rational Zeros: Time to Put on Our Detective Hats

Now comes the fun part: finding the actual rational zeros from our list of possibilities. We'll use a combination of techniques, but one of the most efficient is synthetic division. Synthetic division is like a shortcut for polynomial long division, allowing us to quickly test if a potential zero is a real zero. It's a streamlined process that helps us divide a polynomial by a linear factor of the form (x - c), where 'c' is the potential zero we're testing. If the remainder after synthetic division is zero, then we know that 'c' is indeed a zero of the polynomial.

Let's start by trying some of the simpler possibilities. We can begin with +1 or -1, as these are often good starting points. It's like picking the low-hanging fruit first. We perform synthetic division with 1 and find that the remainder is not zero, so 1 is not a zero. Let's try -5, though. This time, the remainder is zero! Hooray! This means that -5 is a rational zero of f(x). The quotient we obtain from the synthetic division is 15x² - 68x + 9. This is a quadratic equation, which is much easier to deal with than our original cubic polynomial.

Synthetic division isn't just a trick; it's a powerful tool that allows us to reduce the degree of the polynomial. By finding one zero, we've essentially transformed our problem from solving a cubic equation to solving a quadratic equation. This significantly simplifies the process of finding the remaining zeros. Furthermore, synthetic division provides us with the coefficients of the quotient polynomial, which is crucial for factoring the original polynomial. It's like a puzzle piece that helps us complete the bigger picture. So, let's continue to use this technique to unravel the remaining secrets of our polynomial function.

Now, we have a quadratic 15x² - 68x + 9 = 0. We can solve this using the quadratic formula or by factoring. The quadratic formula is a universal tool for solving quadratic equations, while factoring is often a quicker method when applicable. Both techniques rely on understanding the relationship between the coefficients of the quadratic equation and its roots. The quadratic formula, in particular, is a powerful tool that can handle any quadratic equation, regardless of whether it can be easily factored. It guarantees that we will find the solutions, even if they are complex or irrational numbers.

Factoring this quadratic, we get (3x - 1)(5x - 9) = 0. Setting each factor to zero, we find the remaining rational zeros: x = 1/3 and x = 9/5. So, we've found all three rational zeros of our cubic polynomial! We have successfully navigated the world of possible rational zeros and emerged victorious with the actual rational zeros. But our journey isn't over yet. The final step is to factor the original polynomial completely, which will solidify our understanding of its structure and behavior.

Factoring f(x): Putting the Pieces Together

Finally, let's factor f(x) completely. Since we know the zeros are -5, 1/3, and 9/5, we know that the factors are (x + 5), (3x - 1), and (5x - 9). This is a direct consequence of the Factor Theorem, which states that if 'c' is a zero of a polynomial, then (x - c) is a factor of the polynomial. Understanding this theorem is crucial for connecting the zeros of a polynomial to its factored form.

Therefore, we can write f(x) as:

f(x) = 15(x + 5)(x - 1/3)(x - 9/5)

To get rid of the fractions inside the factors, we can multiply the 15 through the last two factors, giving us:

f(x) = (x + 5)(3x - 1)(5x - 9)

And there you have it! We've successfully factored the polynomial f(x). We've not only found the rational zeros but also expressed the polynomial as a product of linear factors. This factored form provides valuable insights into the behavior of the polynomial, such as its roots, its intercepts, and its overall shape. Factoring is a fundamental skill in algebra, and mastering it opens up a world of possibilities for solving equations, simplifying expressions, and understanding mathematical relationships.

So, to recap, we've navigated the world of the polynomial f(x) = 15x³ + 107x² - 247x + 45. We used the Rational Root Theorem to list possible rational zeros, synthetic division to find the actual rational zeros, and the Factor Theorem to factor the polynomial completely. The journey of decoding a polynomial involves a combination of theorems, techniques, and careful analysis. Each step builds upon the previous one, leading us to a deeper understanding of the function's behavior. By mastering these skills, we can confidently tackle a wide range of polynomial problems and appreciate the beauty and power of algebraic concepts. Keep practicing, keep exploring, and keep unlocking the secrets of mathematics!

In conclusion:

• (a) Possible rational zeros: ±1, ±3, ±5, ±9, ±15, ±45, ±1/3, ±5/3, ±1/5, ±3/5, ±9/5
• (b) Rational zeros: -5, 1/3, 9/5
• (c) Factored form: f(x) = (x + 5)(3x - 1)(5x - 9)