remainder thm

2014-11-11 5:16 am

1. when x^3+2x-5 is divided by a polynomial p(x),the quotient and he remainder are x+3 and 13x+1 respectively.
(a) find the polynomial p(x)
(b) find the quotient and the remainder of -2x^3-19x+12x^2+7)/p(x)

2.Let f(x) be the polynomial, its given that the quotient of f(x)/(x-3) is 2x^2+10x-1,and f(3)=4
(a) find the polynomial f(x).
(b) find the remainder when f(x) is divided by 1-2x.

3. let f(x) =ax^3+bx^2-18x+3 and g(x) =ax^3+9x^2+bx-9.when f(x) and g(x) are divided by 2x-1 ,the corresponding remainders are equal.When f(x) is divided by x-2,the remainder is -5��
(a)find the values of a and b.
(b) find the value of k such that the remainder of [f(x)+kg(x)]/(x+4) is 0.

回答 (1)

Eddie

2014-11-14 7:34 pm

✔ 最佳答案

1. (a) (x^3+2x-5)/p(x) = (x+3) … (13x+1)
x^3+2x-5 = (x+3) p(x) + (13x+1)
p(x) = [(x^3+2x-5) - (13x+1)] / (x+3) = (x^3-11x-6)/(x+3) = x^2-3x-2
(b) (-2x^3-19x+12x^2+7)/p(x)
= (-2x^3+12x^2-19x+7)/(x^2-3x-2)
= (-2x+6) ….(-5x+19)

2. (a) f(x)/(x-3) = 2x^2+10x-1 … 4
f(x) = (2x^2+10x-1)(x-3) + 4 = 2x^3+10x^2-x-6x^2-30x+3+4 = 2x^3+4x^2-31x+7
(b) f(1/2)= 2(1/2)^3+4(1/2)^2-31(1/2)+7 = 23.25

3. (a) f(x) = ax^3+bx^2-18x+3, g(x) =ax^3+9x^2+bx-9
When both f(x) and g(x) are divided by 2x-1,
f(1/2) = a(1/2)^3+b(1/2)^2-18(1/2)+3 = a/8+b/4-6
g(1/2) = a(1/2)^3+9(1/2)^2+b(1/2)-9 = a/8+b/2-27/4
f(1/2) = g(1/2)
a/8+b/4-6 = a/8+b/2-27/4
b/2-b/4=27/4-6
b/4 = 3/4
b=3
f(x) = ax^3+3x^2-18x+3
When f(x) is divided by x-2,
f(2) = a(2)^3+3(2)^2-18(2)+3 = -5
8a-21 = -5
8a = 16
a = 2
(b) f(x) = 2x^3+3x^2-18x+3, g(x) =2x^3+9x^2+3x-9
f(x)+kg(x) = 2x^3+3x^2-18x+3 + k(2x^3+9x^2+3x-9)= 2(1+k)x^3+3(1+3k)x^2+3(-6+k)x+3(1-3k)
f(-4)+kg(-4) = 2(1+k)(-4)^3+3(1+3k)(-4)^2+3(-6+k)(-4)+3(1-3k)
= -128-128k+48+144k+72-12k+3-9k
= -5-5k
-5-5k = 0
k=-1

👍 0 👎 0