Maths 問題

(a) Find the quotient when 5x^3 +12x^2 − 9x − 7 is divided by x^2 + 2x − 3 .

5x +2
---------------------------------
x^2 + 2x − 3 )5x^3 +12x^2 − 9x − 7
5x^2 +10x^2 - 15x
-----------------------
2x^2 +6x - 7
2x^2 +4x - 6
-----------------
2x - 1

The remainder = 2x -1,

the quotient is 5x + 2 (ans)

(b) Let g(x) = (5x^3 +12x^2 − 9x − 7) − (ax + b) , where a and b are constants.
It is given that g(x) is divisible by x^2 + 2x − 3 .
(i) since the remainder is 2x - 1
2x - 1 = ax + b

a = 2, b = -1 (ans)

(ii) Solve the equation g(x) = 0 .
(5x^3 +12x^2 − 9x − 7) − (2x - 1)
g(x) = 5x^3 +12x^2 − 11x − 6
Factorize
(x^2 +2x -3)(5x +2)
(x +3) (x – 1) (5x +2) = 0

x - -3, x = -1, x = -0.4 (ans)

Question 2:
Scale A:
M=log4E
E = 4^M
E = (2^2)^M = 2^2M

Scale B:
N=log8E
E = 8^N

E = (2^3)^N = 2^3N
E = 2^2M = 2^3N
2M = 3N
2M = 3(6.4)
M = 3(6.4)/2
M = 3(3.2) = 9.6

the magnitude of the explosion on Scale A is 9.6