Question

The function below has at least one rational root. Find the y-intercept and use the rational roots theorem to find all rational roots. Fill in the sign table and sketch a graph below. Your graph must accurately cross all known intercepts.
f(x) = 2x^3 + 4x^2 - 8x - 16
Identify all real roots. Use commas to separate.

Answer 1

To find the y-intercept, we set x = 0 and evaluate f(x):

f(0) = 2(0)^3 + 4(0)^2 - 8(0) - 16
= 0 + 0 - 0 - 16
= -16

Therefore, the y-intercept is -16.

To identify the rational roots, we can use the rational roots theorem. According to the theorem, the rational roots of a polynomial function can be expressed as p/q, where p is a factor of the constant term (-16) and q is a factor of the leading coefficient (2).

The factors of -16 are: ±1, ±2, ±4, ±8, ±16
The factors of 2 are: ±1, ±2

Therefore, the possible rational roots are: ±1/1, ±2/1, ±4/1, ±8/1, ±16/1, ±1/2, ±2/2 = ±1/2.

To find the real roots, we can perform synthetic division or use a graphing calculator. The real roots are:
x = -2, x = -2, x = 2(sqrt(2))

Here is the sign table:

```
x < -2 | f(x) > 0
-2 < x < 2 | f(x) < 0
x > 2 | f(x) > 0
```

The graph of the function f(x) = 2x^3 + 4x^2 - 8x - 16 would show a positive y-intercept of -16 and would cross the x-axis at x = -2 and x = 2(sqrt(2)) (approximately 2.83).