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
Identify all real roots
To find the y-intercept, we substitute x = 0 into the function:
f(0) = 2(0)^3 + 4(0)^2 - 8(0) - 16
f(0) = 0 + 0 - 0 - 16
f(0) = -16
Therefore, the y-intercept is -16.
To use the Rational Roots Theorem, we identify the coefficients of the function:
a = 2, b = 4, c = -8, d = -16
The Rational Roots Theorem states that if a rational number p/q is a root of the polynomial function, then p is a factor of d (in this case, -16) and q is a factor of a (in this case, 2).
We find the factors of -16 and 2:
Factors of -16: ±1, ±2, ±4, ±8, ±16
Factors of 2: ±1, ±2
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 these fractions, we get:
±1, ±2, ±4, ±8, ±16,
±1/2, ±1, ±2, ±4, ±8
Now, we substitute each of these rational roots into the function and check if the result is equal to zero or not.
f(1) = 2(1)^3 + 4(1)^2 - 8(1) - 16
f(1) = 2 + 4 - 8 - 16
f(1) = -18
f(-1) = 2(-1)^3 + 4(-1)^2 - 8(-1) - 16
f(-1) = -2 + 4 + 8 - 16
f(-1) = -6
f(2) = 2(2)^3 + 4(2)^2 - 8(2) - 16
f(2) = 16 + 16 - 16 - 16
f(2) = 0
Therefore, the rational roots of the function f(x) = 2x^3 + 4x^2 - 8x - 16 are x = 2.
As for the real roots, there may be additional real roots that are not rational. To find these roots, one can use numerical methods such as graphing or using the Newton-Raphson method.