Question

For the polynomial function f(x) = x5-x4-9x3 + 9x2, find the zeros. Then determine the multiplicity at each zero and state whether the graph displays the behavior of a touch or a cross at each intercept.

A.) x = -3, cross; x = 0, cross; x = 1, cross; x = 3, cross

B.) x = -3, cross; x = 0, cross; x = 1, cross; x = 3, touch

C.) x = -3, cross; x = 0, touch; x = 1, cross; x = 3, cross

D.) x = -3, cross; x = 0, cross; x = 1, touch; x = 3, touch

Answer 1

C.) x = -3, cross; x = 0, touch; x = 1, cross; x = 3, cross

Explanation:

You want the zeros and their multiplicity of f(x) = x^5 -x^4 -9x^3 +9x^2, along with a description of whether the graph crosses or touches the x-axis at each zero.

Factors

The equation can be factored as ...

f(x) = x^2(x^3 -x^2 -9x +9)

The cubic can be factored as ...

x^3 -x^2 -9x +9 = x^2(x -1) -9(x -1) = (x^2 -9)(x -1)

and the quadratic factor can be factored as the difference of square. The complete factorization is then ...

f(x) = x^2(x -1)(x -3)(x +3)

Zeros

The zeros are the values of x that make the factors zero:

• x = 0, multiplicity 2
• x = -3, multiplicity 1
• x = 1, multiplicity 1
• x = 3, multiplicity 1

Touch/Cross

When the multiplicity of a zero is even, the graph touches the x-axis. Here, that means the graph will touch at x=0, where the multiplicity of the zero is 2.

When multiplicity is odd, the graph will cross the x-axis. The graph crosses at the zeros -3, -1, +3.

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Additional comment

As soon as you recognize that the x^2 factor means the multiplicity of x=0 is even, you can identify the correct answer choice: the only one that says "touch" for x=0.