To find the remainder of a polynomial when divided by another polynomial, we can use the Remainder Theorem. For our polynomial f(x) = 4x³ + 20x + 50, we want to divide it by x³.
When we divide a polynomial of degree n by a polynomial of degree m, if m < n, the remainder will be a polynomial of degree less than m. In this case, since we are dividing by x³, which is of degree 3, the remainder will be a polynomial of degree less than 3. This means the remainder can be expressed in the form:
- R(x) = ax² + bx + c
Here a, b, and c are constants we need to determine. However, since our polynomial f(x) = 4x³ + 20x + 50 is already of degree 3, we can observe that the term 4x³ will be completely divisible by x³.
This leads us to write:
- f(x) = (4x³) + (20x + 50)
When we divide the polynomial, the 4x³ term cancels out entirely, and we are left with the remainder:
- R(x) = 20x + 50
Thus, the remainder when f(x) is divided by x³ is:
- 20x + 50
So, we conclude that the result is:
- Remainder: 20x + 50