Write a polynomial function that has the zeros 2, -2, and -1 and has a leading coefficient of 1. Then graph the function to show that 2, -2, and -1 are solutions.
Multiply this out:
(x-2)(x+2)(x+1) = (x^2-4)(x+1)
= x^3 +x^2 -4x -4
To find a polynomial function with the given zeros, we can use the fact that if a number is a zero of a polynomial function, then its corresponding factor is (x - zero). In this case, the zeros are 2, -2, and -1, so the corresponding factors are (x - 2), (x + 2), and (x + 1).
To find the polynomial function, we can multiply these factors together:
(x - 2)(x + 2)(x + 1)
Expanding this expression, we get:
(x^2 - 4)(x + 1)
Now, distribute the remaining factor:
(x^2 - 4x + x - 4)
Combining like terms:
(x^2 - 3x - 4)
This is the polynomial function that has the zeros 2, -2, and -1 and has a leading coefficient of 1.
To graph the function and show that the zeros are solutions, we can plot the points (2, 0), (-2, 0), and (-1, 0) on a coordinate plane. Here's what the graph would look like:
```
|
|
|
---|---
| o (2, 0)
| \
| \
| \
| o (-1, 0)
| o (-2, 0)
|
|___________
```
Notice that for each of the points (2, 0), (-2, 0), and (-1, 0), the y-coordinate is indeed 0, indicating that these points lie on the x-axis. Therefore, 2, -2, and -1 are solutions to the given polynomial function.