# F.4 Maths(Polynomial)

When a polynomial g(x) is divided by x-1, the remainder is 5. When g(x) is divided by 2x+1, the remainder is 2.Find the remainder when g(x) is divided by 2x2-x-1.

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• 最佳解答

When a polynomial g(x) is divided by x-1, the remainder is 5. When g(x) is divided by 2x+1, the remainder is 2.Find the remainder when g(x) is divided by 2x2-x-1.

By remainder throrem, when g(x) is divided by x-1, the remainder

= g(1)

So g(1) = 5 ............. (1)

By remainder throrem, when g(x) is divided by 2x+1, the remainder

= g(-1/2)

So g(-1/2) = 2 .......... (2)

As the power of 2x²-x-1 is 2, so the remainder of g(x) divided by 2x²-x-1 is of max. power of 1.

Let g(x) = (2x²-x-1)Q + R

Where Q(x) is the quotient and Rx + C is the remainder

g(x) = (2x²-x-1)Q(x) + Rx + C

g(x) = (x-1)(2x+1)Q(x) + Rx + C ........ (*)

Substitute x=1 into (*)

g(1) = (1-1)(2*1+1)Q + R(1) + C

g(1) = R + C

5 = R + C ............. by (1)

C = 5 - R ............. (3)

Substitute x=-1/2 into (*)

g(-1/2) = (-1/2-1)[2*(-1/2)+1]Q + R(-1/2) + C

g(-1/2) = -R/2 + C

2 = -R/2 + C ............. by (2)

2 = -R/2 + (5-R)

4 = -R + 2(5-R)

4 = -R + 10 - 2R

3R = 6

R = 2

Substitute R=2 into (3)

C = 5-2

C = 3

So the remainder when g(x) is divided by 2x²-x-1

= Rx + C

= 2x + 3

2006-11-29 18:16:12 補充：

小小補充：當 g(x) divided by f(x)，remainder 會是 f(x) 的 (maximum power of x)-1。所以當 g(x) divided by 2x²-x-1，remainder 可以假設是 Rx + c。加上題目先給予的兩個提示 g(1) 和 g(-1/2) 就可以用兩條方程解兩個 variable。