Question

The value of \((p-a) (p-b) (p-c) ........... (p-z)\) is ______ .

• A. A complex polynomial which starts with P²⁴
• B. Zero
• C. A complex polynomial which starts with p²⁶
• D. A complex polynomial which has several variables including p²⁶ and p²⁴

Answer 1

The Correct Option is D

Let's consider the expression \((p-a) (p-b) (p-c) \ldots (p-z)\).

This expression represents a polynomial with 26 factors, each of the form \((p-x)\), where \(x\) is a letter in the sequence from \(a\) to \(z\).

The expression is a product of linear factors. Since every factor is distinct (assuming \(p\) is different from all these letters), it does not trivially evaluate to zero.

To determine the degree of this polynomial, we count the number of factors:

• Each factor contributes 1 to the degree.
• There are 26 distinct factors.

Thus, the degree of the polynomial is 26, indicating the highest power of \(p\) is \(p^{26}\).

Moreover, the expansion of this polynomial will involve various lower degree terms based on combinations of constants \(a, b, c, \ldots, z\). Hence, we will have several cross terms with varying powers of \(p\), including \(p^{24}\) alongside \(p^{26}\).

Therefore, the correct description of the polynomial considering the provided options is: A complex polynomial which has several variables including \(p^{26}\) and \(p^{24}\).