How to Factor a 3 Degree Polynomial
Polynomial equations are a fundamental part of algebra, and factoring them is a crucial skill for solving various mathematical problems. One common type of polynomial is the 3 degree polynomial, also known as a cubic equation. In this article, we will discuss the steps and techniques for factoring a 3 degree polynomial.
Understanding the Basics
Before diving into the process of factoring a 3 degree polynomial, it is essential to understand the basic structure of such equations. A 3 degree polynomial has the general form of ax^3 + bx^2 + cx + d, where a, b, c, and d are constants, and a is not equal to zero. The goal of factoring is to express the polynomial as a product of linear or quadratic factors.
Step 1: Check for Rational Roots
The first step in factoring a 3 degree polynomial is to check for rational roots. Rational roots are the roots that can be expressed as a fraction, where the numerator is a factor of the constant term (d) and the denominator is a factor of the leading coefficient (a). This can be done using the Rational Root Theorem.
Step 2: Find a Rational Root
Once you have identified the possible rational roots, the next step is to find a rational root by substituting each possible root into the polynomial and checking if the result is zero. If a rational root is found, it can be used to factor the polynomial.
Step 3: Factor by Grouping
If a rational root is found, you can factor the polynomial by grouping. To do this, divide the polynomial into two groups, one containing the terms with the highest degree and the other containing the remaining terms. Then, factor out the greatest common factor (GCF) from each group and look for a common factor between the two groups. This common factor can be used to factor the polynomial further.
Step 4: Factor by Synthetic Division
If no rational root is found, you can use synthetic division to test for a root. If a root is found, you can factor the polynomial by synthetic division. This involves dividing the polynomial by the linear factor (x – root) and obtaining a quadratic quotient. The quadratic quotient can then be factored using standard factoring techniques.
Step 5: Factor by Completing the Square
If the polynomial cannot be factored using the above methods, you may need to use the completing the square method. This involves manipulating the polynomial to create a perfect square trinomial, which can then be factored as a product of two binomials.
Step 6: Use the Quadratic Formula
If the polynomial is still not factorable, you can use the quadratic formula to find the roots of the quadratic quotient obtained from synthetic division. The roots can then be used to express the polynomial as a product of linear factors.
Conclusion
Factoring a 3 degree polynomial can be a challenging task, but by following these steps and techniques, you can successfully factor most cubic equations. Practice and familiarity with these methods will help you become more proficient in factoring 3 degree polynomials.
