Short Answer 
The polynomial function f(x) = x³ + 4x² Р4x Р16 can be factored as (x Р2)(x + 2)(x + 4), resulting in roots at x = -2, -4, and 2. These roots indicate the points of intersection with the x-axis, helping identify the correct graph for the function.
Step 1: Analyze the Polynomial
We have the polynomial function, f(x) = x³ + 4x² Р4x Р16. To identify the correct graph for this function, we need to first understand its structure and behavior. Specifically, we will focus on finding its roots, which will indicate where the function intersects the x-axis.
Step 2: Factor the Polynomial
To find the roots of the polynomial, we will factor it. The factorization process goes as follows:
- First rewritten as: f(x) = x²(x + 4) Р4(x + 4)
- This simplifies to: (x² Р4)(x + 4)
- Further factor to: (x – 2)(x + 2)(x + 4)
Now we can set f(x) = 0 to find the roots.
Step 3: Identify the Roots and the Correct Graph
The computed roots from the factored form are x = -2, x = -4, and x = 2. These roots indicate the points where the curve intersects the x-axis, specifically at:
- (-2, 0)
- (-4, 0)
- (2, 0)
Based on these intersections, we can determine that the correct graph for this polynomial function is the one shown in the bottom left of the provided options.