Rational Functions

2015-01-30 6:41 am

1. Find the G.C.D. and L.C.M. of x-2, x²-2x and x³+2x.

2. If the G.C.D. and L.C.M. of x³+10x²+25x and a polynomial P(x) are x²+5x and x^4 +10x³+25x² respectively, find P(x).

3. Let f(x) =2x²+5x-3,g(x)=2x³+x²-13x+6 and h(x)=x²+x-6.
(a) Find the minimum value of f(x).
(b) It is known g(x) is divisible by 2x-1. Factorize g(x) and find the L.C.M. of g(x) and h(x).
(c) Let q(x) = 1/(x²+x-6) - 3/(2x³+x²-13x+6).
(ci) If q(x) = A/f(x) where A is a constant, find A.
(cii) Helen claims that as q(x) = A/f(x), if the minimum value of f(x) is P, then the maximum value of q(x) should be A/P. Agree or not? Explain.

回答 (1)

50418129

2015-01-30 4:13 pm

✔ 最佳答案

(1)

x - 2 = x - 2
x² - 2x = x(x - 2)
x³ + 2x = x(x² + 2)
G.C.D. = 1
L.C.M. = x(x - 2)(x² + 2)

(2)

(x³ + 10x² + 25x) [P(x)] = (x² + 5x)(x^4 +10x³ + 25x²)
x(x + 5)² [P(x)] = x³(x + 5)³
P(x) = x²(x + 5) = x³ + 5x²

(3)

f(x) = 2x² + 5x - 3 = 2(x² + 2.5x) - 3 = 2(x + 1.25)² - 6.125
thus, the minimum value of f(x) is -6.125 or -49/8

g(x) = 2x³ + x² - 13x + 6 = (2x - 1)(x² + x - 6) = (2x - 1)(x + 3)(x - 2)
h(x) = x² + x - 6 = (x + 3)(x - 2)
L.C.M. = (2x - 1)(x + 3)(x - 2)

q(x)
= 1/(x² + x - 6) - 3/(2x³ + x² - 13x + 6)
= 1/[(x + 3)(x - 2)] - 3/[(2x - 1)(x + 3)(x - 2)]
= [(2x - 1) - 3]/[(2x - 1)(x + 3)(x - 2)]
= 2(x - 2)/[(2x - 1)(x + 3)(x - 2)]
= 2/[(2x - 1)(x + 3)] = 2/f(x)
thus, A = 2

do not agree, see what happen when x = -3 and x = 1/2.

參考: knowledge

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