Solving Cubic Equations: A Comprehensive Guide to Finding the Roots of x3 - 6x2 11x - 6 0

When faced with complex polynomial equations, breaking them down into simpler components can provide a clear path to understanding and solving. In this article, we will delve into the process of solving the cubic equation (x^3 - 6x^2 11x - 6 0) step by step.

Understanding the Equation

The given equation is a cubic polynomial, which can be written as:

x3 - 6x2 11x - 6 0

Our goal is to find the values of x that satisfy this equation.

Using the Rational Root Theorem

The Rational Root Theorem states that any rational solution to the polynomial equation (a_nx^n a_{n-1}x^{n-1} cdots a_1x a_0 0) is a factor of the constant term a_0 divided by a factor of the leading coefficient a_n. For our equation, the constant term is -6, and the leading coefficient is 1. Therefore, the possible rational roots are:

1 -1 2 -2 3 -3 6 -6

We will test these potential roots to see which, if any, satisfy the equation.

Testing Potential Roots

Let's test x 1:

[1^3 - 6(1)^2 11(1) - 6 1 - 6 11 - 6 0]

This confirms that x 1 is indeed a root.

Synthetic Division

Since x 1 is a root, we can perform synthetic division to divide the polynomial by x - 1. The synthetic division process is as follows:

11-611-6 11-56 -------------------- 1-5 60

This yields the quotient:

x2 - 5x 6

Factoring the Quadratic

Next, we need to factor the quadratic polynomial x2 - 5x 6:

[x^2 - 5x 6 (x - 2)(x - 3)]

Combining the Factors

Now we can write the original polynomial as:

[x^3 - 6x^2 11x - 6 (x - 1)(x - 2)(x - 3) 0]

Finding the Roots

Setting each factor to zero gives us the roots:

[x - 1 0 quad Rightarrow quad x 1]

[x - 2 0 quad Rightarrow quad x 2]

[x - 3 0 quad Rightarrow quad x 3]

Therefore, the values of x in the equation x3 - 6x2 11x - 6 0 are:

1, 2, 3

This provides a comprehensive solution to the given cubic equation, demonstrating the step-by-step process of solving such equations using the Rational Root Theorem, synthetic division, and factoring techniques.

Additional Approach

As an alternative method, we can also use algebraic manipulation. For the given equation:

x3 - 6x2 11x - 6 0

We can group and rearrange terms to identify a simpler pattern:

[x3 - 3x2 - 3x2 11x - 6 0]

[-3x2 11x - 6 0]

We notice that this can be factored into two simpler polynomials:

[(x - 1)(x2 - 5x 6) x - 1(x - 2)(x - 3) 0]

Thus, the roots are:

x 1, 2, 3

Conclusion

Whether using the Rational Root Theorem, synthetic division, or factoring quadratic equations, the roots of the given cubic equation are x 1, 2, 3. This thorough guide demonstrates the systematic approach needed to solve complex polynomial equations efficiently.