If on division of a non-zero polynomial p(x) by a polynomial g(x), the remainder 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 remainder of the division of \( p(x) \) by \( g(x) \) is zero.

### Step-by-Step Solution:

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

2. **Condition of Remainder**:
According to the problem, the remainder \( r(x) \) is zero:
\[ p(x) = g(x) \cdot q(x) \]
This indicates that \( g(x) \) is a factor of \( p(x) \).

3. **Degree of Polynomials**:
The degree of a polynomial is the highest power of \( x \) in that polynomial. Let:
- \( \text{deg}(p(x)) = n \) (degree of \( p(x) \))
- \( \text{deg}(g(x)) = m \) (degree of \( g(x) \))
- \( \text{deg}(q(x)) = k \) (degree of \( q(x) \))

4. **Degree Relation**:
From the equation \( p(x) = g(x) \cdot q(x) \), we can derive the relationship between the degrees:
\[ \text{deg}(p(x)) = \text{deg}(g(x)) + \text{deg}(q(x)) \]
This implies:
\[ n = m + k \]

5. **Conclusion on Degrees**:
Since \( g(x) \) is a factor of \( p(x) \), the degree of \( g(x) \) must be less than or equal to the degree of \( p(x) \):
\[ m \leq n \]

### Final Relation:
Thus, the relation between the degrees of \( p(x) \) and \( g(x) \) when the remainder is zero is:
\[ \text{deg}(g(x)) \leq \text{deg}(p(x)) \]