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for the polynomial, list each real zero and its multiplicity. determine whether the graph crosses or touches the x-axis at each x-intercept. f(x)=1/2x^4(x^2-5)(x+1)

To find the real zeros (also known as x-intercepts) of the polynomial and determine whether the graph crosses or touches the x-axis at each x-intercept, we need to follow these steps:

Step 1: Factorize the polynomial and rewrite it in factored form.
f(x) = 1/2x^4(x^2-5)(x+1)

Step 2: Identify each factor and set it equal to zero to find the real zeros.
Setting each factor equal to zero:

1/2x^4 = 0
This factor has a zero at x = 0. Since the power of x is even, the graph touches the x-axis at x = 0.

x^2 - 5 = 0
This factor can be solved using the square root property:
x^2 = 5
x = ±√5
So, this factor has zeros at x = √5 and x = -√5. Since the power of x is even, the graph touches the x-axis at x = √5 and x = -√5.

x+1 = 0
This factor has a zero at x = -1. Since the power of x is odd, the graph crosses the x-axis at x = -1.

Step 3: Determine the multiplicity of each zero.
The multiplicity of a zero refers to the number of times a factor occurs as a zero.

In this case, the multiplicity for each zero is as follows:
- The zero at x = 0 has a multiplicity of 4 since it is a factor with a power of 4.
- The zeros at x = √5 and x = -√5 have a multiplicity of 2 since they are factors with a power of 2.
- The zero at x = -1 has a multiplicity of 1 since it is a factor with a power of 1.

So, to summarize:
- The real zero x = 0 has a multiplicity of 4, and the graph touches the x-axis at this point.
- The real zeros x = √5 and x = -√5 have a multiplicity of 2, and the graph touches the x-axis at these points.
- The real zero x = -1 has a multiplicity of 1, and the graph crosses the x-axis at this point.