Step-by-step explanation:
According to factor theorem, if x − c is a factor of f(x), then f(c) = 0.
Conversely, if f(c) = 0, then x −c is a factor of f(x).
a) If x − 3 is a factor of f(x), then f(3) will be 0.
f(3) = 4(3)³ − 12(3)² + 2(3) − 6
f(3) = 0
b) Having shown that x − 3 is a factor of f(x), we can use grouping or long division to factor f(x):
f(x) = 4x³ − 12x² + 2x − 6
f(x) = 4x² (x − 3) + 2 (x − 3)
f(x) = (4x² + 2) (x − 3)
Since 4x² + 2 has only imaginary zeros, 3 is the only real root.
