Solving Polynomials can be tricky especially when they are of a higher order. Below I will outline the steps you can take in order to find all factors of a cubic polynomial and how to write out your answers.
Step 1: Look at the coefficient of the term without a variable attached to it (or the term with an xⁿ where n=0) The coefficient of that term is where you will find ideas of factors to try. Begin by determining the factors of this number and then try inserting them into the equation. If the equation equals zero, you have found a factor of the equation.
Step 2: Once you have your factor in the form of (x – a), where a is the factor found in step 1, you can then proceed to find the remaining factors. There are two different but equally effective ways to do this: by long division or equating coefficients.
Long Division: Simply divide the original equation by the factor found in step one. If done correctly, you should end up with a quadratic function with 0 remainder.
Equating Coefficients:
This method can be a bit more difficult but can save time if you are not comfortable with dividing polynomials. Create an equation with the original expression on one side of an equal sign and on the other side have the factor you found in step 1, in the form (x – a), and multiply that by a general quadratic of the form (Ax2 + Bx + C) so that it looks like this:
(x-a)(Ax2 + Bx + C)
At this point it becomes a bit intuitive as you need to understand where each term in your original expression comes from when you expand the opposite side. An example would be for the term with the x0 attached to it is made from multiplying terms from your found factor and the general quadratic that do not have any variables attached.
From (x-a)(Ax2 + Bx + C) we can deduce that when expanded the only term with no variable attached would be -aC. From there we make -aC equal to the term from the cubic what has no variable and then solve for C, as we have already found what a is in step 1.
After that simply repeat this process for terms with x and x2 in order to solve for A and B. If done correctly the quadratic you get will be the same as the quadratic found through long division.
Step 3: After find your quadratic it’s time to find it’s factors. This can be done by many methods but the most general way would be to use the quadratic formula. These factors you find will be the remaining factors of the original cubic given at the beginning of the problem.
Step 4: Now it’s time to put it all together. We found one factor in step one, a, and in step 3 we found the remaining factors, let them be b and c. Depending on what the question asks, this may be all you needed to do however some questions will ask you write a final statement showing the original expression set equal to it’s factored form. A general example would be:
Ax3 + Bx2 +Cx+D=(x-a)(x-b)(x-c)
If done correctly, expanding the right hand side of the above equation will give you the cubic you started with. It’s a good idea to do this as a way of checking to see if your final answer is correct.
This article was written for you by Troy, one of the tutors with SchoolTutoring Academy.