a) P(x) = x- 2x^2 + 3x^5 + x^4 + x
= 3x^5 + x^4 - 2x^2 + ( x + x )
= 3x^5 + x^4 - 2x^2 + 2x
Q(x) = 3 - 2x - 2x^2 + x^4 - 3x^6 - x^4 + 4x^2
= -3x^6 + ( x^4 - x^4 ) + ( -2x^2 + 4x^2 ) - 2x + 3
-3x^6 + 2x^2 -2x + 3
b) P(x) + Q(x) = ( x - 2x^2 + 3x^5 + x^4 + x ) + ( 3 - 2x - 2x^2 + x^4 -3x^6 - x^4 + 4x^2 )
= x- 2x^2 + 3x^5 + x^4 + x + 3 - 2x - 2x^2 + x^4 - 3x^6 - x^4 + 4x^2
= -3x^6 + 3x^5 + ( x^4 + x^4 - x^4 ) + (-2x^2 - 2x^2 + 4x^2 ) + ( x+x - 2x ) + 3
= -3x^6 + 3x^5 + x^4 + 3
c) Q(x) - P(x) = ( 3 - 2x - 2x^2 + x^4 -3x^6 - x^4 + 4x^2 ) - ( x- 2x^2 + 3x^5 + x^4 + x )
= 3- 2x - 2x^2 + x^4 - 3x^6 - x^4 + 4x^2 - x + 2x^2 - 3x^5 - x^4 - x
= -3x^6 - 3x^5 + ( x^4 - x^4 - x^4 ) + ( -2x^2 + 4x^2 + 2x^2 ) + ( - 2x - x - x ) + 3
-3x^6 -3x^5 - x^4 + 4x^2 - 4x + 3