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 Polynomial Division**:
When dividing 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. **Given Condition**:
According to the problem, the quotient \( q(x) \) is zero. This means:
\[ p(x) = r(x) \]

3. **Degree of Remainder**:
From polynomial division, we know that the degree of the remainder \( r(x) \) must be less than the degree of the divisor \( g(x) \). Therefore, we can write:
\[ \text{degree}(r(x)) < \text{degree}(g(x)) \]

4. **Relating Degrees**:
Since \( p(x) = r(x) \), it follows that:
\[ \text{degree}(p(x)) = \text{degree}(r(x)) \]
Thus, we can combine this with the previous inequality:
\[ \text{degree}(p(x)) < \text{degree}(g(x)) \]

5. **Conclusion**:
The relationship between the degrees of the polynomials is:
\[ \text{degree}(p(x)) < \text{degree}(g(x)) \]

### Final Answer:
If on division of a polynomial \( p(x) \) by a polynomial \( g(x) \), the quotient is zero, then the degree of \( p(x) \) is less than the degree of \( g(x) \).

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