Factors of Polynomials, Part 1

Many problems in engineering and science involve finding the zeroes of a polynomial. This means finding the values of x such that the polynomial is zero. But let’s review what a polynomial is.

A polynomial is anything that can be put in the form:

\[
{a}_{n}{x}^{n}\hspace{0.33em}{+}\hspace{0.33em}{a}_{{n}{-}{1}}{x}^{{n}{-}{1}}\hspace{0.33em}{+}\hspace{0.33em}{a}_{{n}{-}{2}}{x}^{{n}{-}{2}}\hspace{0.33em}{+}\hspace{0.33em}\cdots\hspace{0.33em}{+}\hspace{0.33em}{a}_{1}{x}\hspace{0.33em}{+}\hspace{0.33em}{a}_{0}
\]

where n is a positive integer and the a‘s in front of the x‘s are any real numbers. The numbers in front of the x‘s are called the coefficients. For this post I will only be looking at polynomials with integer (positive or negative) coefficients and polynomials where the first coefficient is 1.

Some examples of polynomials are:

\[
\begin{array}{l}
{{4}{x}^{100}\hspace{0.33em}{-}\hspace{0.33em}{2}{x}^{50}\hspace{0.33em}{+}\hspace{0.33em}{3}{x}^{7}\hspace{0.33em}{-}\hspace{0.33em}{2}}\\
{{x}^{3}\hspace{0.33em}{+}\hspace{0.33em}{7}{x}^{2}\hspace{0.33em}{-}\hspace{0.33em}{4}{x}\hspace{0.33em}{+}\hspace{0.33em}{6}}\\
{{x}\hspace{0.33em}{+}\hspace{0.33em}{6}}\\
{5}
\end{array}
\]

Notice that all the decreasing powers of x do not have to be present. Also, note that numbers by themselves are polynomials as n in this case is 0 and anything to the 0 power is 1.

Now to find the zeroes of these things often requires us to factor the polynomial. That is, change the form of the polynomial t0 several things multiplied together:

\[
{(}{x}\hspace{0.33em}{-}\hspace{0.33em}{a}{)(}{x}\hspace{0.33em}{-}\hspace{0.33em}{b}{)(}{x}\hspace{0.33em}{-}\hspace{0.33em}{c}{)}\hspace{0.33em}\cdots
\]

And we want to do this because of the Null Factor Law. Please see my post about this law but it means that once a polynomial is factored, the values of x that make each factor 0, make the whole polynomial 0.

As an example, it is not obvious what values of x make x² -11x + 30 equal to 0. But if you knew that this polynomial is also equal to (x – 5)(x – 6), then you can immediately see that x = 5 and x = 6 are the zeroes of these factors and are therefore the zeroes of the polynomial.

Now it is not always easy to factor polynomials, but the next few posts will talk about some methods to help do this.

Factoring

Factors are things multiplied together. So numbers can have many different factors. For example, 12 can be seen as 6 × 2, or 3 × 4, or even 2 × 2 × 3. Notice that in the first two sets of factors, the 6 and the 4 can be further broken down to be 2 × 3 and 2 × 2, but the only factors of 2 are 2 × 1 and for 3, 3 × 1. (I am limiting myself to integer factors here because if other types of numbers are allowed, a number has an infinite number of factors).

So numbers like 2 and 3 that only have themselves and 1 as factors are called prime numbers. Other prime numbers are 5, 7, 11, 13, 17, 19, 23. Notice that the only even prime number is 2 as all other even numbers have 2 as a possible factor.

The process of finding the Lowest Common Denominator (LCD) of two or more fractions is done by factoring the denominators into its prime factors. This post is about factoring. I will then use this new skill to find the LCD in my next post.

Now factoring is easier if you know the times tables, commonly up to and including the 12 times tables. When a textbook says to factor a number, it usually means to get the number down to factors of just prime numbers like we did for 12 = 2 × 2 × 3.

Let’s try some:

9 = 3 × 3,  16 = 4 × 4 = 2 × 2 × 2 × 2,  20 = 4 × 5 = 2 × 2 × 5

So generally you just find any two factors of a number then break those factors down further until all that is left are prime numbers. When asked to do this by hand, the numbers are generally small, say less than 144 (12 × 12). Bigger numbers take more work and are usually done by computer programs. By the way, now that you know the factors of 12, you should be able to immediately write down the factors of 144:

144 = 12 × 12 = 2 × 2 × 3 × 2 × 2 × 3

Some more examples:

35 = 5 × 7,   40 = 4 × 10 = 2 × 2 × 2 × 5

100 = 10 × 10 = 2 × 5 × 2 × 5,  60 = 5 × 12 = 5 × 2 × 2 × 3

To find the LCD of two or more denominators, the first step will be to factor each denominator. I will do this in the next post.

Solving factored equations

In my last post, you saw a technique to solve equations when one side consists of factors (things multiplied together) and the other side is zero. Generally, if you have any number of factors with unknowns in them, the only way that the equation can be solved is by setting each factor to zero and solving for the unknown.

That is, if you have say four unknown numbers which multiplied together equal zero: abcd = 0, the only way that this can be true is if any of the unknown numbers are zero.

Let’s have a more complex example. Consider

(x – 4)(x + 7)(x-3)(x+5) = 0

Here are four expressions multiplied together, each with an unknown number x. This can be solved by setting each factor to zero as you can only get zero through multiplication if any of the things multiplied are zero themselves:

x – 4 = 0  ⇒  x = 4

x + 7 = 0  ⇒  x = -7

x – 3 = 0  ⇒  x = 3

x + 5 = 0  ⇒  x = -5

So there are four solutions to this equation.

This is in fact a technique frequently used to solve equations with powers of x

But there are many ways to factor and those skills will be covered in future posts.