The comprehension of polynomials is important throughout multiple tasks because it holds a large position in other academic subjects, improves work performance, and allows mathematical calculations to be simpler and quicker, making daily and complex situations easier.

13. Let f(x) = k be a polynomial in Z2[2]. (5 marks) k= 0 Prove by mathematical induction that for all n EN, [f(2)2 = 22 k=0

Please do it with MATHEMATICA and upload the MATHEMATICA file(.nb)

(a) Solve the differential equation:

     Plot multiple solutions of the differential equation with values of constants

     c–-2.-1.0,1,2 in a single graph

(b) 

     Find the value of y(4) 

(c) Plot the numerical solution of the differential equation for 0  

(6] 8. Let V = P(R) be the vector space of polynomials of degree less than or equal to 2 with usual addition and scalar multiplication. Let W = R+ be the vector space of positive real numbers with vector addition defined by r + s = rs for all r,s EW, and scalar multiplication defined by cr = pl for all TEW,CER For the linear transformation T: V W where T(1+2) = 2, ,T(x2) = 1, T(1 – 1) = find T(a + box + cra).