Final answer:
To divide the polynomial x^3 - 64 by x - 4, we first recognize that the numerator is a difference of cubes and factor it. The resulting factors cancel out the denominator, giving a quotient of x^2 + 4x + 16.
Step-by-step explanation:
The student is asking how to divide a polynomial f(x) = x^3 - 64 by another polynomial g(x) = x - 4. This involves using polynomial long division or applying the factor theorem, since the divisor is of the form x - a, where a is a root of the polynomial f(x) if f(a) = 0.
In this case, 4 is indeed a root of x^3 - 64 since (4)^3 - 64 = 0.
As for the division, we can factor f(x) using the difference of cubes:
- x^3 - 64 = (x - 4)(x^2 + 4x + 16)
Now, we divide (x - 4)(x^2 + 4x + 16) by x - 4, and the x - 4 factors cancel out:
- (x^3 - 64) / (x - 4) = x^2 + 4x + 16