Question

余式定理的一个问题a、b、c是3个不同的实数,f(x)是实系数多项式,已知：f(x)除以(x-a)余a,f(x)除以(x

Answer 1

(x-b)(x-c)f(a)/(a-b)(a-c)+(x-a)(x-c)f(b)/(b-a)(b-c)+(x-a)(x-b)f(c)/(c-a)(c-b)

=(x-b)(x-c)a/(a-b)(a-c)+(x-a)(x-c)b/(b-a)(b-c)+(x-a)(x-b)c/(c-a)(c-b)

=[(x-b)(x-c)(b-c)a-(x-a)(x-c)(a-c)b+(x-a)(x-b)(a-b)c]/(a-b)(a-c)(b-c)

=[(x²-bx-cx+bc)(b-c)a-(x²-ax-cx+ac)(a-c)b+(x²-ax-bx+ab)(a-b)c]/(a-b)(a-c)(b-c)

=[(-bx-cx)(b-c)a-(-ax-cx)(a-c)b+(-ax-bx)(a-b)c]/(a-b)(a-c)(b-c)

=[a(c²-b²)+b(a²-c²)+c(b²-a²)]x/(a-b)(a-c)(b-c)