(中4數)多項式/除數

(1)

設當 f(x) 除以 (x + 1)(x - 2) 的商式為 q(x)，餘式為 ax + b，則
f(x) = (x + 1)(x - 2)q(x) + ax + b
∵ f(-1) = -3 及 f(2) = 3，則
(-1 + 1)(-1 - 2)q(-1) + a(-1) + b = -3 及 (2 + 1)(2 - 2)q(2) + a(2) + b = 3
a - b = 3 及 2a + b = 3
解聯立方程得 a = 2 及 b = -1
∴ 餘式為 2x - 1

(2)(a)

f(1) = g(1)
5(1)³ - 2(1)² + k(1) - 1 = -(1)³ + 2k(1)² - 3(1) + 4
5 - 2 + k - 1 = -1 + 2k - 3 + 4
2 + k = 2k
k = 2

(2)(b)

f(x) - g(x)
= (5x³ - 2x² + 2x - 1) - (-x³ + 4x² - 3x + 4)
= 6x³ - 6x² + 5x - 5
= (x - 1)(6x² + 5)

(3)

search "geogebra" free for download

(x+1)(x-2)=x^2-2x+x-2=x^2-x+2
________
f(x)=(x^2-x+2)Q(x)+ax+b
Index of Remainder is smaller than 2