If on division of a polynomial p(x) by a polynomial g(x), the quotient is zero, what is the relation between the degrees of p(x) and g(x)?

Answer

To solve the problem, we need to analyze the relationship between the degrees of the polynomials \( p(x) \) and \( g(x) \) when the quotient of their division is zero.

### Step-by-Step Solution:

1. **Understanding Division of Polynomials**:
When we divide a polynomial \( p(x) \) by another polynomial \( g(x) \), we can express this as:
\[ p(x) = g(x) \cdot q(x) + r(x) \]
where \( q(x) \) is the quotient and \( r(x) \) is the remainder.

2. **Condition Given**:
The problem states that the quotient \( q(x) \) is zero. This means:
\[ p(x) = r(x) \]
Since \( q(x) = 0 \), the polynomial \( p(x) \) is equal to the remainder \( r(x) \).

3. **Degree of Polynomials**:
The degree of a polynomial is the highest power of \( x \) in the polynomial. Let's denote:
- Degree of \( p(x) \) as \( \deg(p) \)
- Degree of \( g(x) \) as \( \deg(g) \)

4. **Analyzing the Degrees**:
For the division to yield a quotient of zero, the polynomial \( g(x) \) must be of a higher degree than \( p(x) \). This is because if \( \deg(g) \) were less than or equal to \( \deg(p) \), then \( g(x) \) would be able to divide \( p(x) \) to produce a non-zero quotient.

5. **Conclusion**:
Therefore, the relation between the degrees of the polynomials is:
\[ \deg(g) > \deg(p) \]

### Final Answer:
The degree of the polynomial \( g(x) \) must be greater than the degree of the polynomial \( p(x) \).