In previous posts, we have discussed how to solve systems of linear equations. Now, we will talk about a bigger family of equations: the polynomials. Formally, the polynomials are all the equations of the form:
where is some constant (for ) and is a variable. Basically, polynomials are combinations of constants and the positive powers of multiplied by any coefficient.
We say that the highest exponent of a variable in a polynomial is its degree. So for example,
since if we simplify the last equation, the term goes away and we just get which we earlier noted has degree 2.
We say equations with just constants have degree 0. So for example,
Now, some polynomials play a distinctive role in mathematics and so they get special names. Polynomials with degree 2 (i.e. 2 is the highest power of any variable) are called quadratic equations. Here are some examples of quadratic equations:
And while the last equation may not look like our definition of a polynomial, we can see how by subtracting from both sides, we can put it into that form:
Polynomials with degree 3 are called cubic. Here are some examples of those:
In the next few posts, we will discuss two strategies for solving quadratic equations: via the quadratic formula or via factoring.
Find the degree of the following equations:
- , where
Since we know that we may simplify the first term to get Thus, we conclude that the equation is of degree 2.
This equation is of degree 9, since is the highest-degree term in the equation.
This equation is of degree since is the highest-degree term in the equation.