Short Answer 
To find the polynomial from the given zeros -4, 0, and 2, the corresponding factors are (x + 4), x, and (x Р2). Multiplying these factors yields the polynomial function P(x) = x³ + 2x² Р8x, which can be graphed to visualize its behavior at the specified zeros.
Step 1: Identify the Zeros and Factors
Start by recognizing the zeros provided for the polynomial: -4, 0, and 2. These zeros indicate where the polynomial intersects the x-axis. To derive the polynomial factors, convert these zeros into factors of the polynomial as follows:
- (x + 4) for zero -4
- (x – 0) or simply x for zero 0
- (x – 2) for zero 2
Step 2: Formulate the Polynomial Function
To create the polynomial function P(x), multiply the identified factors together. This gives:
- P(x) = (x + 4)(x)(x – 2)
Next, simplify this product step by step:
- P(x) = (x² + 4x)(x Р2)
- P(x) = x³ Р2x² + 4x² Р8x
- P(x) = x³ + 2x² Р8x
Step 3: Graph the Polynomial Function
Once you have the polynomial function P(x), you can plot it to visualize its behavior. Since the zeros are -4, 0, and 2, these points will be where the polynomial intersects the x-axis. By plotting the function, you can see the curve shaped by the polynomial and how it behaves around each zero:
- Understand the curve crosses x-axis at the given zeros
- Observe the general shape and turning points of the graph