Solving X³ + X² = X - 1 Determining The Nature Of Roots

Hey guys! Let's dive into the fascinating world of polynomial equations and explore the roots of the equation x³ + x² = x - 1. This equation, a cubic polynomial, presents a unique challenge in finding its roots. Our journey will involve understanding the nature of roots – whether they are rational, irrational, or complex – and how they relate to the graphical representation of the equation. So, buckle up, and let's get started!

Understanding the Equation and Its Graphical Representation

To kick things off, let's rewrite the equation in the standard polynomial form: x³ + x² - x + 1 = 0. Now, visualizing this equation as a graph can give us some serious insights into its roots. The roots of an equation are essentially the x-intercepts of its graph, the points where the graph crosses the x-axis. Each x-intercept corresponds to a real root of the equation. But here's the catch: not all roots are visible on the graph. You see, complex roots, which involve imaginary numbers, don't show up on the real number plane graph. So, we need to combine our graphical analysis with some algebraic techniques to fully understand the nature of the roots.

When we graph the function y = x³ + x² - x + 1, we'll notice that it intersects the x-axis at only one point. This tells us that there's only one real root. But hold on, a cubic equation should have three roots, right? That's where the complex roots come into play. Since we have one real root, the other two roots must be complex. These complex roots are not visible on the graph because they exist in the complex number plane, not the real number plane that our graph represents. Understanding this interplay between real and complex roots is crucial in solving polynomial equations.

Delving Deeper into Root Types: Rational, Irrational, and Complex

Before we jump to conclusions, let's clarify what we mean by rational, irrational, and complex roots. A rational root is a root that can be expressed as a fraction p/q, where p and q are integers. Think of numbers like 1, -2, or 3/4. An irrational root, on the other hand, is a real root that cannot be expressed as a simple fraction. These roots often involve square roots or other radicals, like √2 or π. Finally, complex roots are numbers that include an imaginary part, denoted by 'i', where i is the square root of -1. Complex roots always come in conjugate pairs (a + bi and a - bi), which is a neat little fact that will be super useful later.

In our case, the graph shows only one x-intercept, indicating one real root. To determine if this root is rational or irrational, we can use the Rational Root Theorem. This theorem helps us list potential rational roots by considering the factors of the constant term (1 in our equation) and the leading coefficient (also 1 in our equation). The potential rational roots are ±1. By plugging these values into the equation, we can check if any of them are actual roots. If none of these potential rational roots work, then our real root must be irrational.

Applying the Rational Root Theorem and Synthetic Division

Let's put the Rational Root Theorem to work. Our potential rational roots are ±1. Plugging x = 1 into the equation x³ + x² - x + 1 = 0, we get 1 + 1 - 1 + 1 = 2, which is not zero. So, 1 is not a root. Now, let's try x = -1: (-1)³ + (-1)² - (-1) + 1 = -1 + 1 + 1 + 1 = 2, which is also not zero. So, -1 isn't a root either. Since neither 1 nor -1 is a root, the real root we found graphically must be irrational. This is a crucial step in narrowing down the possibilities and understanding the nature of the roots.

Now that we know the real root is irrational, let's use synthetic division (or polynomial long division) to divide the polynomial by (x - r), where 'r' is the irrational root. While we don't know the exact value of 'r' yet, this process will help us reduce the cubic equation to a quadratic equation. The quadratic equation will represent the remaining two roots, which we already suspect are complex. This is where things get really interesting, as we'll see how the algebraic manipulation complements our graphical understanding.

Solving for the Roots and Determining Their Nature

After performing synthetic division (which might require numerical methods to approximate the irrational root), we'll end up with a quadratic equation of the form ax² + bx + c = 0. To find the roots of this quadratic equation, we can use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). The discriminant, b² - 4ac, plays a vital role in determining the nature of the roots. If the discriminant is positive, we have two distinct real roots. If it's zero, we have one repeated real root. And if it's negative, we have two complex conjugate roots. Remember our earlier discussion about complex roots coming in pairs? This is where it comes into play.

In our case, since we already know that there's only one real root for the cubic equation, the discriminant of the resulting quadratic equation must be negative. This confirms that the remaining two roots are indeed complex. The quadratic formula will give us these complex roots in the form a + bi and a - bi. This is a classic example of how combining different mathematical tools – graphical analysis, the Rational Root Theorem, synthetic division, and the quadratic formula – allows us to completely solve a polynomial equation and understand the nature of its roots.

Final Verdict: One Rational Root and Two Complex Roots?

So, let's circle back to our initial question: which statement describes the roots of the equation x³ + x² = x - 1? We've established that the equation has one real root and two complex roots. But is the real root rational? We used the Rational Root Theorem to check for potential rational roots, and we found none. Therefore, the real root must be irrational. This means the correct statement is:

One irrational root and two complex roots

This journey through the world of polynomial equations has shown us the power of combining graphical and algebraic methods. Understanding the nature of roots – rational, irrational, and complex – is crucial in solving these equations. And remember, guys, mathematics is like a puzzle; each piece, whether it's a theorem, a formula, or a graph, fits together to reveal the complete picture. Keep exploring, keep questioning, and keep solving!