For the polynomials fx x3x25x2 and gx 3x49x34x23x1 brackets

For the polynomials f(x)= x(3)+x(2)-5x-2 and g(x)= 3x(4)+9x(3)+4x(2)+3x+1. (brackets are powers)

I found the gcd= x(2)+3x+1

Now help me reverse the Euclidean algorithm to express d(x) as d(x)= s(x)f(x)+t(x)g(x). Explicitly state s(x) and t(x).

Solution

For n = 1, f(x) = 6x3 + 4x5 + x2 + 7 and so deg f(x) = 5; For n = 2, f(x) = 4x5 + 4x 5 + x 2 + 7 = x 2 + 7 so deg f(x) = 2; For n = 3, f(x) = 0x 7 + 4x 5 + x2 + 7 and so deg f(x) = 5; For n 4, deg f(x) = 2n+1. Exercise 4: x 3 + 3x2 + 9x + 34 x 3 ) x4 + 7x + 2 x 4 3x3 3x3 + 7x + 2 3x3 9x2 9x2 + 7x + 2 9x2 27x 34x + 2 34x 102 104 Hence the quotient is x3 + 3x2 + 9x + 34 and the remainder is 104. If we just wanted the reminder it would be much easier to use the Remainder Theorem. The remainder is f(3) = 81 + 21 + 2 = 104.

Rather than divide we can first subtract. This will simplify the arithmetic. GCD(a(x), b(x)) = GCD(a(x) b(x), b(x)) = GCD(2x2 29x 48, 8x2 + 30x + 27). = GCD(2x2 29x 48, (8x2 + 30x + 27) 4(2x2 29x 48)) = GCD(2x2 29x 48, 146x + 219) = GCD(2x2 29x 48, 2x + 3) since 146x + 219 = 73(2x + 3). x 16 2x + 3 ) 2x2 29x 48 2x2 + 3x 32x 48 32x 48 0