f(x) = x²(x² + 1)(x + 4)

To find the y-intercept, substitute x=0

f(0) = 0²(0² + 1)(0 + 4)
f(0) = 0

So the y-intercept is (0,0)

The domain of every polynomial function is (-∞,∞), "all real numbers".

To find the zeros, set f(x) = 0 and solve for x

x²(x² + 1)(x + 4) = 0

Use the zero factor principle, that is, set each factor =0

x² = 0;   x² + 1 = 0;   x + 4 = 0
 x = 0;       x² = -1;      x = -4
               x = ±V-1;
               x = ±i;  

So there are four zeros, 2 real ones and two imaginary ones:

They are:

0, -4, i, -i

A zero R will have multiplicity of M if M is the largest positive 
integer such that (x-R)M is a factor of the polynomial.  
This polynomial has the factor x² which is equivalent to (x-0)²,
so the multiplicity of the zero 0 is 2, an even number, so the curve
"bounces off" the x-axis at 0.  The factor (x+4)1 has 
exponent 1, an odd number so the zero -4 has multiplicity 2, an even
number and the graph will cut through the x-axis at -4.


End behavior:

Rule: 

multiply the right side all the way out:

f(x) = x²(x² + 1)(x + 4)
f(x) = x5 + 4x4 + x3 + 4x2

Look at the leading term, the one with the largest exponent, 1x5 

Rules:

1. If its coefficient is positive the curve goes UP on the extreme right.
2. If its coefficient is negative the curve goes DOWN on the extreme right.
3. If the largest exponent (the degree) is even, the extreme left hand behavior
   is the SAME as the right hand behavior.
4. If the exponent (the degree) is odd, the extreme left hand behavior is the
   OPPOSITE of the right hand behavior.

Since  1x5 has a positive coefficient, the extreme right hand
behavior is UP on the right.
Since  1x5 has an ODD exponent, the extreme left hand behavior is
the OPPOSITE, so the extreme left-hand behavior is DOWN of the extreme left.   

Here is the graph:



Notice the on the far right the curve goes UP, and on the far left it goes
DOWN.  Notice that the graph "bounces off" the x axis at the zero 0, and cuts
through the x-axis at the zero -4.

Edwin