Final answer:
To find the zeros of f(x) = x³ - 4x² + 16x - 64, we can identify that x = 4 is a zero by pattern recognition and then factor the cubic equation. Further factorization will lead to the remaining zeros.
Step-by-step explanation:
To find all the zeros of the function f(x) = x³ - 4x² + 16x - 64, we will look for values of x that make the function equal to zero. However, this is not a quadratic equation but a cubic equation. In order to solve cubic equations, we sometimes need to look for patterns or use methods such as synthetic division or factoring, if applicable. In this case, we can factor by grouping or notice a pattern.
Let's observe the pattern in this cubic equation:
Firstly, notice that the constant term is -64, which is -4 raised to the third power.
The coefficient of x is 16, which is 4 times 4.
Looking at the equation we can suspect that (x - 4) might be a factor, as substituting x = 4 into the equation we would have all terms with positive 64 subtracted by 64 leading to zero.
Using x = 4 and synthetic division or factoring, we can find that (x - 4) is indeed a factor. Factoring the cubic polynomial completely, we get:
\(f(x) = (x - 4)(x^2 + ax + 16)\)
We can find the constant a by polynomial division or other factorization methods. Once we factor completely, we can then set each factor equal to zero and solve for x to find all zeros of the function.
If (x - 4) is a known zero, we could also consider possibilities for complex roots since the function has real coefficients. In such cases, we would expect complex roots to be conjugates.