Author: Indigo Curnick
Date: 2025-04-02
Something we need to do all the time in mathematics is solve polynomials. There's many methods for doing this thankfully! In this blog we'll go over methods of solving polynomials by hand, and leave numerical methods for another day.
Factorising by Inspection
If we have a quadratic formula, we can usually factorise by inspection. This is basically a fancy way to say "we can look at it and guess the answer". Consider
We can ask "what numbers add to 2 and times to -8?", and in this case a solution presents itself quite simply as
Thus the roots are
Sometimes we can try several combinations, since expanding is so quick. Consider
We know that when we factorise this, one of the factors will have a
but if you expand this, it won't quite work. A little trial and error will lead you to
Try and solve the following using inspection
Quadratic Formula
Sometimes factorising isn't so simple. Consider the polynomial
Which has solutions
For
So from our example
Try the following using the quadratic formula
Completing the Square
Another method of solving quadratics is completing the square. In this method, we arrange the quadratic into a perfect square form, rather than the more common two brackets form. (You might also notice that this is how the quadratic formula can be derived, though we won't explicitly give that proof).
The goal of completing the square is to get the quadratic in the form of
into the form
Where
So if we take an example of
Try the following out
Comparing Coefficients
So far, we've been solving quadratics only. That's okay, but eventually you will need to solve higher order polynomials. There is a cubic formula, and a fourth order formula, but they're way too tedious to memorise. Also there is no fifth order or higher formula (which we won't prove here). A much better method is to factorise by comparing coefficients.
Now the first trick with this kind of factorisation is you need to know one factor. There's unfortunately no special method for this or anything - you'll just need to use trial and error to find it. Sometimes looking at the polynomial might give you a hint to find a factor.
Let's take the example of
If we expand the RHS out we get
Now we can compare the coefficients
So
At this point, since you have a quadratic, you can easily solve that with one of the methods listed so far (for your convenience, it is
Try the following problems (Hint: all these polynomials have at least one solution which is an integer under 10 😉)
You can find the solutions here