Mastering the Art of Factoring Polynomials- A Comprehensive Guide to Solving Cubic Equations

How to Factor Polynomial with Degree 3

Polynomial factoring is a fundamental skill in algebra that involves expressing a polynomial expression as a product of simpler expressions. One common type of polynomial is the cubic polynomial, which has a degree of 3. Factoring a cubic polynomial can be challenging, but with the right approach, it can be done efficiently. In this article, we will discuss various methods to factor a polynomial with degree 3.

One of the most straightforward methods to factor a cubic polynomial is by using the Rational Root Theorem. This theorem states that if a polynomial has integer coefficients, then any rational root of the polynomial must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. By testing possible rational roots, we can find one or more roots of the polynomial and factor it accordingly.

Here’s a step-by-step guide on how to factor a cubic polynomial using the Rational Root Theorem:

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. Form all possible combinations of these factors to create possible rational roots.
4. Test each possible root by substituting it into the polynomial. If the result is zero, then the root is valid.
5. Once a valid root is found, use polynomial division to divide the original cubic polynomial by the linear factor (x – root).
6. The quotient obtained from the division will be a quadratic polynomial. Factor this quadratic polynomial using standard factoring techniques or the quadratic formula.
7. Combine the linear and quadratic factors to obtain the factored form of the cubic polynomial.

For example, let’s factor the cubic polynomial f(x) = x^3 – 5x^2 + 6x – 6:

1. The constant term is -6, and the leading coefficient is 1.
2. The factors of -6 are ±1, ±2, ±3, and ±6, while the factors of 1 are ±1.
3. Possible rational roots are ±1, ±2, ±3, ±6, ±1/2, ±3/2, ±1/3, and ±2/3.
4. Testing each possible root, we find that x = 1 is a root.
5. Dividing f(x) by (x – 1) gives us the quotient x^2 – 4x + 6.
6. The quadratic polynomial x^2 – 4x + 6 cannot be factored further, so we use the quadratic formula to find its roots: x = (4 ± √(16 – 24)) / 2 = 2 ± √2i.
7. Combining the linear and quadratic factors, we get the factored form of f(x): (x – 1)(x – 2 – √2i)(x – 2 + √2i).

Another method to factor a cubic polynomial is by using the sum or difference of cubes formula. This formula states that a^3 ± b^3 = (a ± b)(a^2 ± ab + b^2). By identifying the appropriate values of a and b in a cubic polynomial, we can apply this formula to factor the polynomial.

For instance, consider the cubic polynomial g(x) = x^3 + 8:

1. Recognize that g(x) can be written as (x)^3 + (2)^3.
2. Apply the sum of cubes formula: x^3 + 8 = (x + 2)(x^2 – 2x + 4).
3. The quadratic factor x^2 – 2x + 4 cannot be factored further, so the factored form of g(x) is (x + 2)(x^2 – 2x + 4).

In conclusion, factoring a polynomial with degree 3 can be achieved using the Rational Root Theorem or the sum or difference of cubes formula. By applying these methods, you can simplify cubic polynomials and solve various algebraic problems.