If `f(x)` and `g(x)` are of degrees 7 and 4 respectively such that `f(x)= g(x) q(x)+ r(x)` then find possible degrees of `q(x)` and `r(x)`.

Answer

To solve the problem, we start with the equation given:

\[ f(x) = g(x) \cdot q(x) + r(x) \]

where:
- \( f(x) \) is a polynomial of degree 7,
- \( g(x) \) is a polynomial of degree 4,
- \( q(x) \) is the quotient,
- \( r(x) \) is the remainder.

### Step 1: Understand the relationship between degrees

According to polynomial division, the degree of the remainder \( r(x) \) must be less than the degree of the divisor \( g(x) \). Therefore, we have:

\[ \text{degree of } r(x) < \text{degree of } g(x) \]

### Step 2: Determine the degree of \( r(x) \)

Since the degree of \( g(x) \) is 4, we can conclude:

\[ \text{degree of } r(x) < 4 \]

This means the possible degrees of \( r(x) \) can be:

- 0 (constant polynomial)
- 1 (linear polynomial)
- 2 (quadratic polynomial)
- 3 (cubic polynomial)

Thus, the possible degrees of \( r(x) \) are:

\[ 0, 1, 2, \text{ or } 3 \]

### Step 3: Determine the degree of \( q(x) \)

Next, we need to find the degree of \( q(x) \). The degree of \( f(x) \) is 7, and we can express the relationship of the degrees as follows:

\[ \text{degree of } f(x) = \text{degree of } g(x) + \text{degree of } q(x) \]

Substituting the known values:

\[ 7 = 4 + \text{degree of } q(x) \]

From this, we can solve for the degree of \( q(x) \):

\[ \text{degree of } q(x) = 7 - 4 = 3 \]

### Conclusion

Thus, the possible degrees of \( q(x) \) and \( r(x) \) are:

- **Possible degree of \( q(x) \)**: 3
- **Possible degrees of \( r(x) \)**: 0, 1, 2, or 3