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 - 20x^2 + 100x)

determine the zero(s), if they exist.

the zero(s) is/are
(type integers or decimals. use a comma to separate answers as needed.)

Explanation:

Factor out the greatest common factor

To find the zeros of the polynomial function, we set \(f(x) = 0\):

$$ x^3 - 20x^2 + 100x = 0 $$

We factor out the greatest common factor, which is \(x\):

$$ x(x^2 - 20x + 100) = 0 $$

Factor the quadratic expression

Using the Perfect Square Trinomials knowledge point, we recognize that the quadratic expression inside the parentheses is a perfect square:

$$ x^2 - 20x + 100 = (x - 10)^2 $$

Thus, the fully factored form of the polynomial function is:

$$ f(x) = x(x - 10)^2 $$

Determine the zeros of the function

Using the Multiplicity of Zeros knowledge point, we set each factor to zero to find the roots:

$$ x = 0 \quad \text{or} \quad (x - 10)^2 = 0 \implies x = 10 $$

The zeros of the function are \(0\) and \(10\).

Answer:

Determine the zero(s), if they exist.

The zero(s) is/are <blank>0, 10</blank>
(Type integers or decimals. Use a comma to separate answers as needed.)