Question

Consider the following polynomial function f(x)=(x+1)(x-1)(x-3)List each real zero of f according to the behavior of the graph at the x-axis near that zero if there is more than one answer, separate them with commas.Zero(s)where the graph crosses the x-axis:Zero(s) where the graph touches, but does not cross the x-axis:Find the y-intercept of the graph f:

Answer 1

We are given the following polynomial function

$f(x)=(x+1)(x-1)(x-3)$

We are asked to list the zeros of the function according to the behavior of the graph.

Zeros where the graph crosses the x-axis:

The graph crosses the x-axis at these points

$\begin{gathered} x+1=0 \\ x=-1 \\ x-1=0 \\ x=1 \\ x-3=0 \\ x=3 \end{gathered}$

So, there are three zeros where the graph crosses the x-axis.

-1, 1, 3

Zero(s) where the graph touches but does not cross the x-axis:

There is no such zero there touches the x-axis but does not cross it.

Find the y-intercept of the graph:

The y-intercept is the point where the graph crosses the y-axis.

Substitute x = 0 into the function to find the y-intercept.

$\begin{gathered} f(x)=(x+1)(x-1)(x-3) \\ f(x)=(0+1)(0-1)(0-3) \\ f(x)=(1)(-1)(-3) \\ f(x)=3 \end{gathered}$

So, the y-intercept of the graph is 3