Final answer:
Linear transformation T applied on the polynomial 4 - 3x is calculated by expressing 4 - 3x as a linear combination of given transform results, yielding the answer -2.5 + 1.25x.
Step-by-step explanation:
The student is asking for the result of the linear transformation T on the polynomial 4 - 3x. We are given that T(1 + 5x) = -4 + 3x and T(5 + 24x) = -2 - 2x. To find T(4 - 3x), we can express the polynomial 4 - 3x as a linear combination of 1 + 5x and 5 + 24x, for which we already know the transformations. Through some algebra, we find that 4 - 3x can be expressed as:
½(1 + 5x) + ¼(5 + 24x).
Since linear transformations are compatible with addition and scalar multiplication, we have:
T(4 - 3x) = T(½(1 + 5x)) + T(¼(5 + 24x))
We know that:
T(1 + 5x) = -4 + 3x
T(5 + 24x) = -2 - 2x
Therefore:
T(½(1 + 5x)) = ½(-4 + 3x) = -2 + ½(3x)
T(¼(5 + 24x)) = ¼(-2 - 2x) = -0.5 - 0.5x
Adding these together, we get:
T(4 - 3x) = (-2 + ½(3x)) + (-0.5 - 0.5x) = -2.5 + ½(2.5x) = -2.5 + 1.25x.