F.4 maths function

[匿名]

2013-02-18 10:22 pm

Let f(x)=(x-a)(x-b)(x+1)-3,where a and b are positive integers with a>b.It is given
that f(1)=1

a(i) prove that (a-1)(b-1)=2
(ii) write down the values of a and b

b. let g(x)=x^3-6x^2-2x+7. Using the results of a(ii),find f(x)-g(x).Hence find the
exact values of all roots of the equation f(x) =g(x)

回答 (1)

用戶10012

2013-02-19 12:46 am

✔ 最佳答案

a(i) Put x = 1
f(1) = (1 - a)(1 - b)(1 + 1) - 3 = 1
(1 - a)(1 - b)(2) = 4
(1 - a)(1 - b) = 2
[ - (a - 1)][ - (b - 1)] = 2.
(a - 1)(b - 1) = 2.
a(ii) Since a and b are positive integers, (a - 1) and (b - 1) are also positive integers. For product of 2 positive integers to be equals to 2, the 2 integers can only be 2 x 1.
That is (a - 1) = 2 or 1, a = 3 or 2
(b - 1) = 2 or 1, b = 3 or 2.
Since a > b , so a = 3 and b = 2.
so f(x) = (x - 3)(x - 2)(x + 1) - 3
= x^3 - 4x^2 + x + 3.
(b)
f(x) - g(x) = 2x^2 + 3x - 4
f(x) = g(x) means f(x) - g(x) = 0
so 2x^2 + 3x - 4 = 0
x = [- 3 +/- sqrt 41]/4
(2x

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