多項式.....40分

(1)(a) 設f(x) = x^3 + x^2 + mx + 3
f(-3) = -27 + 9 – 3m + 3 = 0
3m = -15
m = -5
(b) f(x)/(x+3) = x^2 – 2x + 1 = (x – 1)^2
f(x) = (x-1)^2(x + 3)
(2)(a) 設f(x) = x^3 + Cx^2 – x – C
f(4) = 64 + 16C – 4 – C = 0
60 + 15C = 0
C = -4
(b) f(x) = x^3 – 4x^2 – x + 4
f(x) / (x – 4) = x^2 – 1 = (x + 1)(x – 1)
f(x) = (x – 1)(x + 1)(x – 4)
(3) 設f(x) = 2x^30 – ax + 11
f(1) = 2 – a + 11 = 0
a = 13
(4)設f(x) = x^2 + 3x – k – 3
f(k) = k^2 + 3k – k – 3 = 0
k^2 + 2k – 3 = 0
(k + 3)(k – 1) = 0
k = -3或 k = 1
(5) p(x)=6x^3-25x^2+3x+4
(a)(i) p(-1/2) = -6/8 – 25/4 – 3/2 + 4 = -4.5
(2x + 1)不是因式
(ii) p(3/2) = 162/8 – 225/4 + 9/2 + 4 = -27.5
(2x – 3)不是因式
(iii) p(-1/3) = -6/27 – 25/9 – 1 + 4 = 0
(3x + 1) 是因式
(b) p(x) / (3x + 1) = 2x^2 – 9x + 4 = (2x – 1)(x – 4)
p(x) = (2x – 1)(x – 4)(3x + 1)

1)
x+3 = 0
x = -3
先代 x = -3
f(-3) = (-3)^3 + (-3)^2 -3m + 3 = 0 (因式 所以系根)
m = -5
再用長除法.
x^3 +x^2 - 5x +3 除 x+3
就會知道
x^3 +x^2 - 5x +3
等於
(x+3)(x^2-2x+1)
= (x+3)(x-1)(x-1)
= (x+3)(x-1)^2.

其他都系甘計.
希望幫到你.