`px^(3) + qx^(2)+rx+ s` is said to be a cubic polynomial, if ____.

Options

A. `s ne 0`
B. `r ne 0`
C. `q ne 0`
D. `p ne 0`

Answer

To determine when the expression \( px^3 + qx^2 + rx + s \) is classified as a cubic polynomial, we need to analyze the structure of the polynomial.

### Step-by-Step Solution:

1. **Identify the General Form of a Cubic Polynomial**:
A cubic polynomial is generally expressed in the form:
\[ ax^3 + bx^2 + cx + d \]
where \( a, b, c, \) and \( d \) are constants, and \( a \) (the coefficient of \( x^3 \)) must be non-zero.

2. **Identify the Highest Degree**:
In the polynomial \( px^3 + qx^2 + rx + s \), the term with the highest degree is \( px^3 \). The degree of a polynomial is determined by the highest power of \( x \) present in the expression.

3. **Condition for Cubic Polynomial**:
For the polynomial to be classified as cubic, the coefficient \( p \) (the coefficient of \( x^3 \)) must not be zero. If \( p = 0 \), the term \( px^3 \) disappears, and the highest degree of the polynomial would then be determined by the next highest power, which is \( qx^2 \). This would make it a quadratic polynomial instead.

4. **Conclusion**:
Therefore, the condition for \( px^3 + qx^2 + rx + s \) to be a cubic polynomial is:
\[ p \neq 0 \]

### Final Answer:
The expression \( px^3 + qx^2 + rx + s \) is said to be a cubic polynomial if \( p \neq 0 \).