Problem 1. We are going to multiply the two polynomials A(x) = 5 − 3x and...

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 three different 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 the values of C(x) at the three roots.

(iii) Plug 1, i, −1 into C(x) = a + bx + cx2 to form three simultaneous equations with three unknowns.

(iv) Solve for a, b, c.

(c) (i) Evaluate A(x) and B(x) at the four (real) 4th roots of unity 1, i, −1, −i.

(ii) Multiply the values at the four 4th roots to form the values 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).

Expert Solution

Colby Messinger answered 1 year ago

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