Question

escribing the end behavior of a function
complete the statements that describe the x-intercepts of the graph of \\( f(x) = x^5 - 8x^4 + 16x^3 \\).

the graph touches, but does not cross, the x-axis at \\( x = \square \\).

the graph of the function crosses the x-axis at \\( x = \square \\).

Explanation:

Step 1: Factor the function

Start with \( f(x) = x^5 - 8x^4 + 16x^3 \). Factor out the greatest common factor, \( x^3 \):
$$ f(x) = x^3(x^2 - 8x + 16) $$

Step 2: Factor the quadratic

The quadratic \( x^2 - 8x + 16 \) is a perfect square trinomial, since \( 8/2 = 4 \) and \( 4^2 = 16 \). Thus:
$$ x^2 - 8x + 16 = (x - 4)^2 $$

Step 3: Write the fully factored form

Substitute back into \( f(x) \):
$$ f(x) = x^3(x - 4)^2 $$

Step 4: Analyze the roots

• For \( x^3 = 0 \), the root is \( x = 0 \) with multiplicity 3 (odd multiplicity).
• For \( (x - 4)^2 = 0 \), the root is \( x = 4 \) with multiplicity 2 (even multiplicity).

Step 5: Determine end behavior at each root

• A root with odd multiplicity (like \( x = 0 \), multiplicity 3) means the graph crosses the \( x \)-axis at that point.
• A root with even multiplicity (like \( x = 4 \), multiplicity 2) means the graph touches but does not cross the \( x \)-axis at that point (it “bounces” off the axis).

Answer:

The graph touches, but does not cross, the \( x \)-axis at \( x = 4 \).
The graph of the function crosses the \( x \)-axis at \( x = 0 \).