Problem 1. We are going to multiply the two polynomials A(x) = 5− 3x and B(x) = 4 + 2x to produce C(x) = a + bx + cx2 in threedifferent ways. Do this by hand, and show your work.
(a) Multiply A(x) × B(x) algebraically.
(b) (i) Evaluate A and B at the three (real) roots of unity 1,i, −1. (Note that we could use any three values.)
(ii) Multiply the values at the three roots of unity to form thevalues of C(x) at the three roots.
(iii) Plug 1, i, −1 into C(x) = a + bx + cx2 to form threesimultaneous equations with three unknowns.
(iv) Solve for a, b, c.
(c) (i) Evaluate A(x) and B(x) at the four (real) 4th roots ofunity 1, i, −1, −i.
(ii) Multiply the values at the four 4th roots to form thevalues of C(x) at the four 4th roots.
(iii) Create the polynomial D(x) = C(1) + C(i)x + C(−1)x 2 +C(−i)x 3 .
(iv) Evaluate D(x) at the four 4th roots of unity 1, i, −1, −i.(v) Use these values to construct C(x).
