Maths about Polynomials 超急

2006-11-27 4:49 am

Let f(x) = 3x^2 - 6x - 1 and g(x) = A(x-1)(x+1) + B(x-1)^2 + C(x+1)^2. When f(x) and g(x) are divided by x, x+1 and x+2 respectively, both f(x) and g(x) leave the same remainder,
(a) Find the values of A, B and C.
(b) Show that f(x) ≡ g(x).
(c) Solve g(x) = 8.

回答 (1)

myisland8132

2006-11-27 5:06 am

✔ 最佳答案

for f(x)
When f(x) is divided by x, remainder=-1
When f(x) is divided by x+1, remainder=f(-1)=8
When f(x) is divided by x+2, remainder=f(-2)=12+12-1=23
since both f(x) and g(x) leave the same remainder
g(0)=-A+B+C=-1
g(-1)=4B=8
g(-2)=3A+9B+C=23
we get B=2
2-A+C=-1
18+3A+C=23
that is
A=C+3
18+3(C+3)+C=23
4C=-4
C=-1
A=2
(b)
f(x) = 3x^2 - 6x - 1
g(x) = A(x-1)(x+1) + B(x-1)^2 + C(x+1)^2
g(x) = 2(x-1)(x+1) + 2(x-1)^2 - (x+1)^2.
g(x) = 2(x^2-1) + 2(x-1)^2 - (x+1)^2.
g(x) = 2x^2-2 + 2x^2-4x+2 - x^2-2x-1
g(x) = 3x^2 - 6x - 1
f(x) ≡ g(x)
(c)
g(x) = 8
3x^2 - 6x - 1=8
3x^2 - 6x - 9=0
(3x+3)(x-3)=0
x=-1 or x=3

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