# Using Zeros to Graph Polynomials - Loose Definition Needed for Comprehension

I am having a hard time understanding this definition.

The definition states.

"If P is a polynomial function, then "c" is called a zero of P if P(c) = 0. In other words, the zeros of P are the solutions of the polynomial equation P(x) = 0. Note that if P(c) = 0, then the graph of P has an x-intercept at x = c, so the x-intercepts of the graph are the zeros of the function.

Real Zeros of Polynomials

If P is a polynomial and c is a real number, then the following are equivalent:

1) c is a zero of P 2) x= c is a solution of the equation P(x) = 0 3) x-c is a factor of P(x) 4) c is an x-intercept of the graph of P."

Been reading this definition for and 1 hour and I just don't get it. Can someone explain it to me as if I am a 5 year old?

• The definition of an $x$-intercept of a function $f$ is just a place where $f(x)=0$, so that relates the "intercept" part. Also $x=c$ is a solution of an equation if when you plug that value into the equation it is true (i.e. the equality holds). So that relates the "solution" part. The only non-trivial part is that $x-c$ is a factor of $P$, but without getting too technical, in practice you can check this by performing long division of polynomials – user139388 Apr 5 '14 at 21:00

A number $c \in \mathbb{C}$ (I use $\mathbb{C}$ rather than $\mathbb{R}$ to allow for complex zeroes) is called a zero of a polynomial $P(x)$ if and only if $P(c) = 0$. Graphically, if a number $c$ is a zero of $P(x)$, then the point $(c,0)$ is on the graph of $P$, and so it hits the $x$-axis at the point $c$.