Finding Zeroes: F(x) = X^4 - 4x^3 - 4x^2 + 36x - 45

Determining the zeroes, also known as roots, of a polynomial function is a fundamental task in algebra and calculus. In this comprehensive guide, we will explore the process of finding all zeroes of the function f(x) = x^4 - 4x^3 - 4x^2 + 36x - 45. This particular function is a quartic polynomial, meaning it has a degree of four, which implies it can have up to four roots (including complex roots). To effectively tackle this problem, we will employ a combination of techniques including the Rational Root Theorem, synthetic division, and factoring. Let’s dive in and systematically find each zero of this function.

Understanding the Importance of Finding Zeroes

Before we delve into the step-by-step process, it's crucial to understand why finding zeroes is so important. The zeroes of a function are the x-values where the function's graph intersects the x-axis. These points are also known as the roots or solutions of the equation f(x) = 0. Knowing the zeroes of a function provides critical information about its behavior, including where it changes sign, its overall shape, and its long-term trends. In practical applications, finding zeroes can help solve real-world problems in fields such as engineering, physics, economics, and computer science. For instance, in engineering, understanding the roots of a system's characteristic equation can reveal stability criteria. In economics, zeroes might represent equilibrium points. Therefore, mastering the techniques to find zeroes is an invaluable skill.

Step 1: Applying the Rational Root Theorem

The Rational Root Theorem is our first powerful tool in finding potential rational zeroes of the polynomial. This theorem states that if a polynomial has integer coefficients, then any rational zero must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In our function, f(x) = x^4 - 4x^3 - 4x^2 + 36x - 45, the constant term is -45 and the leading coefficient is 1. Thus, the factors of -45 are ±1, ±3, ±5, ±9, ±15, ±45, and the factors of 1 are ±1. The possible rational roots are therefore ±1, ±3, ±5, ±9, ±15, and ±45. This narrows down the infinite possibilities to a manageable list of candidates. To determine which of these are actual roots, we will test them using synthetic division.

Step 2: Testing Potential Roots with Synthetic Division

Now that we have a list of potential rational roots, we use synthetic division to test each one. Synthetic division is an efficient method for dividing a polynomial by a linear factor (x - c), where c is a potential root. If the remainder of the division is zero, then c is indeed a root of the polynomial. Let's start by testing x = 1. Performing synthetic division with 1, we get a non-zero remainder, indicating that 1 is not a root. Next, we test x = -1. Again, synthetic division yields a non-zero remainder, so -1 is not a root either. We continue this process, and when we test x = 3, we find that the remainder is zero. This means that x = 3 is a root of the polynomial, and (x - 3) is a factor. The result of the synthetic division also gives us the quotient, which is a cubic polynomial. This is a significant step because it reduces the degree of the polynomial we need to work with, making it easier to find the remaining roots. After finding one root, the problem becomes more manageable as we deal with lower-degree polynomials.

Step 3: Reducing the Polynomial

After confirming that x = 3 is a root, the synthetic division process provides us with the quotient polynomial, which is x^3 - x^2 - 7x + 15. This cubic polynomial represents the remaining part of the original quartic function after dividing out the factor (x - 3). Now, we need to find the zeroes of this reduced polynomial. We can apply the Rational Root Theorem again, but this time to the cubic polynomial. The possible rational roots are the factors of 15 (±1, ±3, ±5, ±15). We already know that x = 3 is a root of the original polynomial, so we can try these new candidates. Testing x = -3 using synthetic division on the cubic polynomial, we find that the remainder is zero. Therefore, x = -3 is another root. This reduces the polynomial further, making it a quadratic. Finding roots becomes simpler as the degree of the polynomial decreases, allowing for more straightforward methods such as factoring or using the quadratic formula.

Step 4: Solving the Quadratic Equation

Having found two roots (x = 3 and x = -3), we now have a quadratic quotient from the synthetic division, which is x^2 - 4x + 5. To find the remaining roots, we need to solve this quadratic equation. There are several methods to solve a quadratic equation, including factoring, completing the square, and using the quadratic formula. In this case, the quadratic does not factor easily, so we will use the quadratic formula. The quadratic formula is given by: x = [-b ± sqrt(b^2 - 4ac)] / (2a), where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. For our equation, x^2 - 4x + 5 = 0, a = 1, b = -4, and c = 5. Plugging these values into the formula, we get: x = [4 ± sqrt((-4)^2 - 4(1)(5))] / (2(1)) = [4 ± sqrt(16 - 20)] / 2 = [4 ± sqrt(-4)] / 2. The discriminant (b^2 - 4ac) is negative, indicating that the roots are complex numbers.

Step 5: Finding Complex Roots

From the quadratic formula, we obtained x = [4 ± sqrt(-4)] / 2. Since sqrt(-4) = 2i (where i is the imaginary unit, sqrt(-1)), the complex roots are x = (4 ± 2i) / 2, which simplifies to x = 2 ± i. These are the two remaining roots of the original quartic polynomial. Complex roots always come in conjugate pairs for polynomials with real coefficients, which is the case here. The discovery of complex roots completes our search for all the zeroes of the function, highlighting the importance of being comfortable with complex numbers in polynomial algebra.

Conclusion

In summary, we have successfully found all the zeroes of the function f(x) = x^4 - 4x^3 - 4x^2 + 36x - 45. The zeroes are x = 3, x = -3, x = 2 + i, and x = 2 - i. We achieved this by using a combination of the Rational Root Theorem, synthetic division, and the quadratic formula. This comprehensive approach is applicable to many polynomial functions and demonstrates the power of algebraic techniques in solving complex problems. Understanding these methods is crucial for anyone studying algebra, calculus, and related fields. The process we followed here is a general strategy that can be adapted to find the roots of other polynomials, making it a valuable skill for both academic and practical applications. Remember to always consider the Rational Root Theorem as a starting point, use synthetic division to reduce the polynomial's degree, and apply appropriate methods such as factoring or the quadratic formula to find the remaining roots. Finding zeroes is a cornerstone of polynomial analysis and opens the door to deeper understanding and applications.

For further reading and a deeper dive into polynomial functions and their properties, consider visiting a trusted resource like Khan Academy's Algebra II section.