Question:
Let T : p5(x) → p5(x) such that T(1) = 1 and T [x(x-1)(x-k 1)] , k varies from 1 to 5, then what is the value of T(x^3) and T(x^4)? Explain in detail.

Solution:

To find the values of T(x^3) and T(x^4), we need to understand the given function T : p5(x) → p5(x) and the given condition T(1) = 1.

Understanding the Function:
The function T : p5(x) → p5(x) means that the function T maps polynomials of degree at most 5 to polynomials of degree at most 5. In other words, the input and output of the function are polynomials, specifically polynomials of degree 5 or less.

Condition T(1) = 1:
The given condition T(1) = 1 tells us that when the polynomial 1 is input into the function T, the output is also 1. This means that T(1) = 1.

Using T[x(x-1)(x-k 1)], k varies from 1 to 5:
The given expression T[x(x-1)(x-k 1)], where k varies from 1 to 5, represents the action of the function T on the polynomial x(x-1)(x-k 1). We need to find the value of T(x^3) and T(x^4), which can be obtained by substituting x^3 and x^4 respectively into the given expression.

Value of T(x^3):
Substituting x^3 into T[x(x-1)(x-k 1)], we get:
T(x^3) = T[x(x-1)(x-k 1)] = x^3(x^3-1)(x^3-k 1)

Value of T(x^4):
Substituting x^4 into T[x(x-1)(x-k 1)], we get:
T(x^4) = T[x(x-1)(x-k 1)] = x^4(x^4-1)(x^4-k 1)

Summary:
- The function T : p5(x) → p5(x) maps polynomials of degree at most 5 to polynomials of degree at most 5.
- The condition T(1) = 1 means that when the polynomial 1 is input into the function T, the output is also 1.
- Substituting x^3 and x^4 into T[x(x-1)(x-k 1)], we can find the values of T(x^3) and T(x^4) respectively.
- The value of T(x^3) is x^3(x^3-1)(x^3-k 1) and the value of T(x^4) is x^4(x^4-1)(x^4-k 1).

Please note that EduRev is an educational platform and we are here to provide academic assistance. We do not promote any external websites.