In this lesson, our instructor Vincent Selhorst-Jones teaches Roots (Zeros) of Polynomials. This lesson teaches about roots in graphs and factoring, as well as, naïve attempts. The factoring sections goes over how to find roots, check your work, and factor higher degree polynomials. Youll learn why not all polynomials can be factored and why there is a limit on the max number of roots/factors. The lesson ends with four great examples to give you extra practice.
The roots/zeros/x-intercepts of a polynomial are the x-values where the polynomial equals 0.
If you have access to the graph of a function/equation, it is very easy to see where the roots are: where the graph cuts the horizontal axis (the x-intercepts)! Why? Because that's where f(x) = 0 or y=0.
While you can occasionally find the roots to a polynomial by trying to isolate the variable and directly solve for it, that method often fails or is misleading.
To find the roots of a polynomial we need to factor the polynomial: break it into its multiplicative factors. Then we can set each factor to 0 and solve to find the roots.
Factoring can be quite difficult if you're trying to factor a very large or complicated polynomial. There is no procedure that will work for factoring all polynomials.
In general, if we have a quadratic trinomial (something in the form ax2 + bx + c), we can factor it into a pair of linear binomials as
ax2 + bx + c = ( x + ) ( x + ).
Think about what has to go in each blank for it to be equivalent to the polynomial you started with.
Whenever you're factoring polynomials, make sure you check your work! Even on an easy problem, there are ample opportunities to make a mistake. That means you should always try expanding the polynomial (it's fine to do it in your head) to make
sure you factored it correctly.
In general, factoring higher degree polynomials is similar to what we did above. Figure out how you can break down the polynomial into a structure like the above, then ask yourself how you can fill in the blanks.
If you already know a root to a polynomial, it must be one of its factors. For example, if we know x=a is a root, then the polynomial must have a factor of (x−a). This makes factoring the polynomial that much easier.
Not all polynomials can be factored. Sometimes it is impossible to reduce it to smaller factors. In such a case, we call the polynomial irreducible. [Later on, we'll discuss a a hidden type of number we haven't previously explored when we learn
about the complex numbers. These will allow us to factor these supposedly irreducible polynomials. However, for the most part, we won't work with complex numbers so such polynomials will stay irreducible.]
There is a limit to how many roots/factors a polynomial can have. A polynomial of degree n can have, at most, n roots/factors.
We also get information about the possible shape of a polynomial's graph from its degree. A polynomial of degree n can have, at most, n−1 peaks and valleys (formally speaking, relative maximums and minimums).
Roots (Zeros) of Polynomials
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.