高中數學公倍式.工因式問題

qwe091653

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chongwaikei

we suppose that

f(x) = s(x)d(x) , g(x) = t(x)d(x)

observe that , m(x) = s(x)t(x)d(x)= f(x)g(x)/d(x)  and  deg(d(x)) ≦ 3

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by using factorization

we know deg(d(x)) ≠ 1 and deg(m(x)) = 5

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p(x)f(x) + q(x)g(x) = d(x)

p(x)s(x) + q(x)t(x) = 1

the equation looks like the form ax + by = 1, where it is Diophantine Equation
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chongwaikei

if you don't know how to factorize the functions

here is a method

f(x) = x^3 - x^2 - 11 x - 4
      
      = x^3 - 64 - (x^2 - 11x + 60)

      = (x - 4)(x^2 + 4x + 16) - (x - 4)(x + 15)

      = (x - 4)(x^2 + 3x + 1)

g(x) = x^4 + 2x^3 + 5x + 2
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個人習慣用輾轉相除法求公因式，因式分解有時候會無從下手。

補充一下 (D) 選項，

假設有一組 p1 和 q1 滿足 p1*s+q1*t=1，
令 p2 = p1+t，q2 = q1-s，
則 p2*s+q2*t = (p1+t)*s+(q1-s)*t = p1*s+t*s+q1*t-s*t = 1