How to Factor a Third Degree Polynomial
In mathematics, factoring a polynomial is an essential skill that helps in solving various equations and simplifying expressions. A third degree polynomial, also known as a cubic polynomial, is a polynomial of the form ax^3 + bx^2 + cx + d, where a, b, c, and d are real numbers and a is not equal to zero. In this article, we will discuss different methods to factor a third degree polynomial, including the Rational Root Theorem, synthetic division, and factoring by grouping.
1. Rational Root Theorem
The Rational Root Theorem is a fundamental tool used to find the possible rational roots of a polynomial. For a third degree polynomial ax^3 + bx^2 + cx + d, the possible rational roots are the factors of the constant term d divided by the factors of the leading coefficient a. Once we have the list of possible roots, we can use synthetic division to test each of them and find the actual roots.
To apply the Rational Root Theorem, follow these steps:
1. Find the factors of the constant term d.
2. Find the factors of the leading coefficient a.
3. Form the list of possible rational roots by dividing the factors of d by the factors of a.
4. Test each possible root using synthetic division.
2. Synthetic Division
Synthetic division is a method used to divide a polynomial by a linear factor (x – r). If the remainder of the division is zero, then r is a root of the polynomial. To factor a third degree polynomial using synthetic division, we need to find at least one root first. Once we have a root, we can use synthetic division to divide the polynomial by (x – r) and obtain a quadratic polynomial.
To factor a third degree polynomial using synthetic division, follow these steps:
1. Find a root of the polynomial using the Rational Root Theorem and synthetic division.
2. Divide the polynomial by (x – r) using synthetic division.
3. Factor the resulting quadratic polynomial using standard factoring techniques, such as factoring by grouping or the quadratic formula.
3. Factoring by Grouping
Factoring by grouping is a method used to factor a polynomial by grouping terms and then factoring out a common factor. This method can be applied to third degree polynomials when the polynomial has four terms.
To factor a third degree polynomial using factoring by grouping, follow these steps:
1. Group the polynomial into two pairs of two terms.
2. Factor out the greatest common factor (GCF) from each pair.
3. Find a common factor between the two GCFs and factor it out.
4. The remaining polynomial should be a quadratic, which can be factored further using standard techniques.
In conclusion, factoring a third degree polynomial can be achieved using various methods, such as the Rational Root Theorem, synthetic division, and factoring by grouping. By applying these techniques, you can simplify complex expressions and solve cubic equations efficiently.
