Question

Let T: P_3 rightarrow P_3 be the linear transformation satisfying T(1) = 2x^2 + 2, T(x) = 4x - 9, T(x^2) = -4x^2 + x - 2. Find the image of an arbitrary quadratic polynomial ax^2 + bx + C. T(ax^2 + bx + c) =

Answer 1

Answer:

$T(ax^2+bx+c)=(-4a+2c)x^2+(a+4b)x-2a-9b+2c$

Step-by-step explanation:

Let T:$P_3\rightarrow P_3$ be the linear transformation

$T(1)=2x^2+2$

$T(x)=4x-9$

$T(x^2)=-4x^2+x-2$

We have to find the image of an arbitrary quadratic polynomial

$ax^2+bx+c$

We know that linear transformation satisfied the property

$T(ax+by)=a T(x)+b T(y)$

Therefore, $T(ax^2+bx+c= a T(x^2)+b T(x)+ c T(1)$

Substitute all given  values in right place

$T(ax^2+bx+c)=a (-4x^2+x-2)+b(4x-9)+c(2x^2+2)$

$T(ax^2+bx+c)=(-4a+2c)x^2+(a+4b)x-2a-9b+2c$

Hence, the image of an arbitrary quadratic polynomial

$T(ax^2+bx+c)=(-4a+2c)x^2+(a+4b)x-2a-9b+2c$.