The function below has at least one rational root. Find the y-intercept and use the rational roots theorem to find all rational roots.
f(x) = 2x^3 + 4x^2 - 8x - 16
To find the y-intercept, we plug in x = 0 into the equation f(x) = 2x^3 + 4x^2 - 8x - 16:
f(0) = 2(0)^3 + 4(0)^2 - 8(0) - 16
= 0 + 0 - 0 - 16
= -16
Therefore, the y-intercept is -16.
Now, let's use the Rational Roots Theorem to find all rational roots. According to the theorem, any rational root of the polynomial equation f(x) = 2x^3 + 4x^2 - 8x - 16 must be a factor of -16 (the constant term) divided by a factor of 2 (the leading coefficient).
The factors of -16 are ±1, ±2, ±4, ±8, ±16.
The factors of 2 are ±1, ±2.
Combining these factors, the possible rational roots are:
±1/1, ±2/1, ±4/1, ±8/1, ±16/1,
±1/2, ±2/2, ±4/2, ±8/2, ±16/2.
Simplifying, we have the following potential rational roots:
±1, ±2, ±4, ±8, ±16,
±1/2, ±1, ±2, ±4, ±8.
To determine which, if any, of these potential rational roots are actually roots of the equation, we can use polynomial long division, synthetic division, or a calculator's polynomial solver.