Algebra
Polynomials
We have already gone over what equations are, and even one type of them (linear equations). Now, let’s cover a different type of equation: polynomials. A polynomial is a huge umbrella that contains many different types of equations (including linear equations!). The form of a polynomial is:
\(P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0}\) \(a_{n}\) is the coefficient of the variable, \(x\). an can be any number. In each term, x is raised to some exponent, \(n\). \(n\) is always an integer that is at least 0. These terms are separated by addition (or subtraction, if the term after has a negative coefficient). There are two observations that are important. First, that if \(a_{n}\) is 0, then the entire term is multiplied by 0 and is therefore canceled out. Also, if the exponent of \(x\) is 0, then that means that the corresponding term is whatever the coefficient is, because \(x^0\) is always equal to 1. Some examples of polynomials are:
\[f(x)=3x^5+x^4+13x^3−11x+x^2+10\\ g(x)=x+x^2-3\\ h(x)=7x^4−2x−\frac{18}{5}\]Some examples of functions that are not polynomials are:
\[i(x)=\frac{x^2+5x}{x^9-35}\text{ because the entire equation is a fraction}\] \[j(x)=x^3-2x^2+8x-4+3x^{-1}\text{ because there is at least one negative exponent}\] \[k(x)=2x^3+x^\frac{1}{2}\text{ because there is at least one non-integer exponent}\]The constant of a polynomial is the term that has \(x^0\) as it’s variable part. The constant is just a number. For example, in the polynomial
\[f(x)=3x^5+x^4+13x^3−11x^2+10\]the constant is \(10\). In the polynomial
\[g(x)=x+x^2-3\]the constant is \(-3\).
The degree of the polynomial is equal to the highest exponent of the polynomial. For example, in the polynomial
\[f(x)=3x^5+x^4+13x^3−11x^2+10\]the degree is \(5\). In the polynomial
\[g(x)=x+x^2-3\]the degree is \(2\).
As a final note, know that polynomials are usually ordered with the highest exponent on the far left to the lowest exponent (the constant) on the far right. A polynomial doesn’t have to be ordered like that, but that is how they should be ordered in most cases.