# hwk4 - C x at the four 4th roots(iii Create the polynomial...

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Fall 2011 CMSC 451: Homework 4 Clyde Kruskal Due at the start of class Thursday, November 10, 2011. Problem 1. We are going to multiply the two polynomials A ( x ) = 5 - 3 x and B ( x ) = 4+2 x to produce C ( x ) = a + bx + cx 2 in three di±erent 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 + cx 2 to form three simultaneous equations with three unknowns. (iv) Solve for a , b , c . (c) (i) Evaluate A and B 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
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Unformatted text preview: 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 at the four 4th roots of unity 1, i ,-1,-i . (v) Use these values to construct C ( x ). Problem 2. Use the FFT algorithm to evaluate f ( x ) = 8-4 x + 2 x 2 + 3 x 3-5 x 4-4 x 5 + 2 x 6 + x 7 at the eight 8th roots of unity mod 17. You may stop using recursion when evaluating a linear function ( a + bx ), which is easier to do directly. The eight 8th roots of unity mod 17 are 1, 2, 4, 8, 16, 15, 13, 9; it is easier to calculate with 1, 2, 4, 8, -1, -2, -4, -8. Do this by hand, and show your work....
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