Question

Find the zeros for the given polynomial function and give the multiplicity for each zero. State whether the graph
crosses the x-axis or touches the x-axis and turns around at each zero.

f(x)=x^3-40x^2+400x

Determine the zero(s), if they exist.

The zero(s) is/are
(Type integers or decimals. Use a comma to separate answers as needed.)

Answer 1

To find the zeros of the polynomial function \( f(x) = x^3 - 40x^2 + 400x \), we can start by factoring the expression.

1. First, we can factor out the common term from the polynomial:
\[
f(x) = x(x^2 - 40x + 400)
\]

2. Next, we need to factor the quadratic \( x^2 - 40x + 400 \). We can either use the quadratic formula or complete the square. However, we can look for perfect squares first:
\[
x^2 - 40x + 400 = (x - 20)^2
\]
Therefore, we can rewrite the factored polynomial as:
\[
f(x) = x(x - 20)^2
\]

3. Now, to find the zeros, set \( f(x) = 0 \):
\[
x(x - 20)^2 = 0
\]

This gives us:
\[
x = 0 \quad \text{and} \quad (x - 20)^2 = 0 \implies x = 20
\]

4. The zeros of the polynomial are:
- \( x = 0 \) (multiplicity 1)
- \( x = 20 \) (multiplicity 2)

5. Now, we determine if the graph crosses or touches the x-axis at each zero:
- At \( x = 0 \) (multiplicity 1), since the multiplicity is odd, the graph **crosses** the x-axis.
- At \( x = 20 \) (multiplicity 2), since the multiplicity is even, the graph **touches** the x-axis and turns around.

### Summary:
- The zeros are: \( 0, 20 \)
- Multiplicity:
- \( 0 \): multiplicity 1 (crosses the x-axis)
- \( 20 \): multiplicity 2 (touches and turns around at the x-axis)

Thus, the zeros are:
\[
\text{The zero(s) is/are: } 0, 20
\]