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.