Factorising Polynomials of Degree 3 A polynomial of degree 3 is a polynomial with the highest degree of 3. It has the form: $$p(x) = ax^3 + bx^2 + cx + d...
A polynomial of degree 3 is a polynomial with the highest degree of 3. It has the form:
where a, b, c, and d are constants.
Factorising a polynomial of degree 3 requires us to find three distinct factors that multiply together to give the coefficients of the leading and middle terms.
Step 1: Find the middle coefficient (b)
The middle coefficient is the coefficient of the squared term (x^2). In a polynomial of degree 3, the middle coefficient will be 3.
Step 2: Find the coefficients of the leading term (a) and constant term (d)
The coefficient of the leading term is the coefficient of x^3, which is a. The constant term is the coefficient of x + d, which is the leading term multiplied by a constant factor.
Step 3: Identify the three factors
Once we have the coefficients of the leading and middle terms, we can identify the three distinct factors that will multiply together to give these coefficients. These factors are often found by factoring the middle coefficient divided by the leading coefficient, like this:
Example:
Consider the polynomial:
From this, we can identify the following:
a = 3
b = 2
c = -1
d = 4
Therefore, the factors of p(x) are:
Tips for Factoring Polynomials of Degree 3:
Look for the largest factors of the middle coefficient first.
If the factors are not integers, they will likely be complex conjugates.
Use long division to find the factors of the leading term.
Remember that the factors of a polynomial of degree 3 will always be in the form of (ax + b)(cx^2 + dx + e)