Find Zeros: F(x) = 5x³-3x²+3x+2 - Explained!

Alright, guys, let's dive into finding all the zeros of the function f(x) = 5x³ - 3x² + 3x + 2. This is a classic problem in algebra, and we're going to break it down step by step. We'll explore different techniques and strategies to tackle this cubic equation. So, buckle up, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what we're trying to find. The zeros of a function, also known as roots or x-intercepts, are the values of x for which f(x) = 0. In other words, we're looking for the values of x that make the equation 5x³ - 3x² + 3x + 2 = 0 true. For a cubic function like this one, we can expect to find up to three zeros, which can be real or complex numbers.

Why Finding Zeros Matters

Finding the zeros of a function is a fundamental concept in mathematics with wide-ranging applications. Zeros can represent equilibrium points in physics, break-even points in economics, or critical values in engineering. Understanding how to find zeros allows us to analyze and model real-world phenomena more accurately.

The Challenge of Cubic Equations

Unlike quadratic equations, which have a straightforward formula (the quadratic formula), cubic equations don't have a simple, universally applicable formula. This means we often need to rely on a combination of techniques, such as factoring, synthetic division, and numerical methods, to find the zeros.

Factoring Techniques

One of the first approaches to try is factoring. Factoring involves rewriting the polynomial as a product of simpler polynomials. If we can factor the cubic function into linear factors, we can easily find the zeros by setting each factor equal to zero.

Looking for Rational Roots

The Rational Root Theorem can be a helpful tool in identifying potential rational roots. The theorem states that if a polynomial has integer coefficients, then any rational root 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 case, the constant term is 2, and the leading coefficient is 5. Therefore, the possible rational roots are ±1, ±2, ±1/5, and ±2/5.

Testing Potential Roots

We can test these potential roots by plugging them into the function and checking if the result is zero. Let's try x = -2/5:

f(-2/5) = 5(-2/5)³ - 3(-2/5)² + 3(-2/5) + 2 = 5(-8/125) - 3(4/25) - 6/5 + 2 = -8/25 - 12/25 - 30/25 + 50/25 = (-8 - 12 - 30 + 50)/25 = 0/25 = 0

So, x = -2/5 is indeed a root of the function. This means that (x + 2/5) is a factor of the polynomial.

Synthetic Division

Now that we've found one root, we can use synthetic division to divide the cubic polynomial by (x + 2/5) and obtain a quadratic polynomial. Synthetic division is a shortcut method for dividing a polynomial by a linear factor.

Let's perform synthetic division with x = -2/5:

-2/5 | 5 -3 3 2
 | -2 2 -2
 ------------------
 5 -5 5 0

The result of the synthetic division is the quadratic polynomial 5x² - 5x + 5. This means that we can rewrite the original cubic function as:

f(x) = (x + 2/5)(5x² - 5x + 5)

Solving the Quadratic Equation

Now we need to find the zeros of the quadratic equation 5x² - 5x + 5 = 0. We can use the quadratic formula to solve for x:

x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 5, b = -5, and c = 5. Plugging these values into the quadratic formula, we get:

x = (5 ± √((-5)² - 4(5)(5))) / (2(5)) = (5 ± √(25 - 100)) / 10 = (5 ± √(-75)) / 10 = (5 ± 5i√3) / 10 = (1 ± i√3) / 2

So, the two complex roots are x = (1 + i√3) / 2 and x = (1 - i√3) / 2.

The Zeros of the Function

We have found all three zeros of the function f(x) = 5x³ - 3x² + 3x + 2:

1. x = -2/5
2. x = (1 + i√3) / 2
3. x = (1 - i√3) / 2

Therefore, the function has one real root and two complex roots. This completes the solution!

Alternative Methods

While we successfully found the zeros using factoring and the quadratic formula, there are other methods we could have used. Numerical methods, such as Newton's method or the bisection method, can be used to approximate the roots of the equation. These methods are particularly useful when dealing with more complicated functions that are difficult to factor.

Newton's Method

Newton's method is an iterative technique that uses the derivative of the function to find successively better approximations of the roots. The formula for Newton's method is:

x_(n+1) = x_n - f(x_n) / f'(x_n)

where x_n is the current approximation and x_(n+1) is the next approximation. By repeatedly applying this formula, we can converge to a root of the function.

Bisection Method

The bisection method is another iterative technique that involves repeatedly dividing an interval in half and selecting the subinterval that contains a root. This method is guaranteed to converge to a root, but it may be slower than Newton's method.

Tips and Tricks

Here are some tips and tricks to keep in mind when finding the zeros of a function:

• Always start by looking for rational roots using the Rational Root Theorem.
• Use synthetic division to reduce the degree of the polynomial.
• If you encounter a quadratic equation, use the quadratic formula.
• Consider using numerical methods for more complicated functions.
• Double-check your work to avoid errors.

Conclusion

Finding the zeros of a function can be a challenging but rewarding task. By understanding the different techniques and strategies available, you can tackle even the most complicated equations. Remember to start with factoring, use synthetic division, and don't be afraid to try numerical methods when necessary. With practice and perseverance, you'll become a pro at finding zeros!

So, there you have it, guys! We've successfully found all the zeros of the function f(x) = 5x³ - 3x² + 3x + 2. Keep practicing, and you'll become a master of algebra in no time!

Keywords: Zeros of a function, roots, x-intercepts, cubic equation, factoring, Rational Root Theorem, synthetic division, quadratic formula, complex roots, Newton's method, bisection method.

Keep up the great work!