How to Solve a Third Degree Polynomial
Polynomials are a fundamental concept in mathematics, and solving them is an essential skill for anyone studying algebra or calculus. Among the different types of polynomials, the third degree polynomial, also known as a cubic equation, presents a unique challenge. In this article, we will explore various methods to solve a third degree polynomial equation.
1. Factoring by grouping
The first method to solve a third degree polynomial is factoring by grouping. This technique involves rewriting the polynomial as a product of two binomials. To do this, we need to find two binomials whose product equals the original polynomial. The process can be summarized as follows:
1. Group the terms of the polynomial in pairs.
2. Factor out the greatest common factor (GCF) from each pair.
3. Factor out the GCF from the entire polynomial.
For example, consider the polynomial \(x^3 – 5x^2 + 4x – 20\). We can group the terms as follows:
\((x^3 – 5x^2) + (4x – 20)\)
Now, factor out the GCF from each pair:
\(x^2(x – 5) + 4(x – 5)\)
Finally, factor out the GCF from the entire polynomial:
\((x – 5)(x^2 + 4)\)
Now we can solve the equation by setting each factor equal to zero:
\(x – 5 = 0\) or \(x^2 + 4 = 0\)
The solutions for the first equation are \(x = 5\), and for the second equation, there are no real solutions since \(x^2 + 4\) is always positive.
2. Rational Root Theorem
The Rational Root Theorem is another method to solve a third degree polynomial. It states that if a polynomial has rational roots, then those roots must be of the form \(\pm\frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.
To use the Rational Root Theorem, follow these steps:
1. Identify the constant term and the leading coefficient of the polynomial.
2. Find all the factors of the constant term and the leading coefficient.
3. Test each combination of factors to see if they are roots of the polynomial.
For example, consider the polynomial \(x^3 – 6x^2 + 11x – 6\). The constant term is \(-6\) and the leading coefficient is \(1\). The factors of \(-6\) are \(\pm1, \pm2, \pm3, \pm6\), and the factors of \(1\) are \(\pm1\). By testing each combination, we find that \(x = 1\) is a root of the polynomial.
3. Cardano’s method
Cardano’s method is a more complex method to solve cubic equations. It involves using complex numbers and algebraic manipulation. The steps for Cardano’s method are as follows:
1. Convert the cubic equation to the form \(x^3 + px + q = 0\).
2. Find two numbers \(u\) and \(v\) such that \(u^3 + v^3 = -p\) and \(uv = \frac{q}{3}\).
3. Express the solution as \(x = u + v\).
4. Solve for \(u\) and \(v\) using the quadratic formula.
This method can be quite lengthy and is generally used when other methods are not applicable or when dealing with complex cubic equations.
In conclusion, solving a third degree polynomial can be done using various methods such as factoring by grouping, the Rational Root Theorem, and Cardano’s method. Each method has its advantages and limitations, and the choice of method depends on the specific polynomial and the desired level of accuracy.
