Maths!!請幫幫忙!

2008-09-24 7:29 am

When a polynomial P(x) is divided by a polynomial S(x), the quotient is 2x^2 - 3x - 2 and the remainder is 12.
(a) If P(x) ≡(x+p)(2x+q)S(x) + r, where p, q, r are constants and q>p. Find the values of p, q and r.
(b) It is known that P(x) is a polynomial of degree 3, and is in the form 4x^3 + 4x^2 + ax + b. Find
(i) the values of a and b and
(ii) S(x).
(c) Find the remainder when P(x) is divisible by x + 2.
.
Please in detail.
.
A trader bought some paraffin for $500. He paid $x for each litre of paraffin.
(a) Find, in terms of x, an expressoin for the number of litres he bought.
(b) Due to a leak, he lost 3 litres of paraffin. He sold the remainder of the paraffin for $1 per litre more than he paid for it. Write down an expression, in terms of x, for the sum of money he received.
(c) He made a profit of $20
(i) Write down an equation in x to represent this information, and show that it reduces to 3x^2 + 23X - 500 =0.
(ii) Solve the equation 3x^2 + 23X - 500 =0, giving both answers correct to one decimal place.
(d) Find, correct to the nearest whole number, how many litres of paraffin he sold.

回答 (1)

用戶10012

2008-09-24 3:38 pm

✔ 最佳答案

1.
(a)2x^2 - 3x -2 = (2x + 1)(x -2). That is
P(x) = (2x +1)(x -2)S(x) + 12. Therefore, p = -2, q = 1 and r = 12.
(b)
4x^3 + 4x^2 + ax + b = (2x +1)(x -2)S(x) + 12.
Put x = 2
32 + 16 + 2a + b = 12
2a + b = - 36.........................(1)
Put x = -1/2
-1/2 + 1 -a/2 + b = 12
-1 + 2 - a + 2b = 24
2b - a = 23..........................(2)
From (1) and (2) we get a = -19 and b = 2.
So P(x) = 4x^3 + 4x^2 - 19x + 2 = (2x +1)(x -2)S(x) + 12.
Comparing coefficient of x^3 and the constant term on both side, we get
S(x) = 2x + 5.
(c)
Remainder = P(-2) = -32 + 16 + 38 + 2 = 24.
(II)
(a)Number of litres bought = 500/x.
(b)Quantity sold = 500/x - 3.
Price per litre = x + 1.
Amount received = (500/x - 3)(x + 1).
(c)
Profit = (500/x - 3)(x + 1) - 500 = 20
(500 - 3x)(x +1)/x = 520
500x + 500 - 3x^2 - 3x = 520x, that is
3x^2 + 23x - 500 = 0
(ii) x = [-23 +/- sqrt(23 x 23 + 4 x 3 x 500)]/6 = (-23 +/- 80.8)/6 = 9.6. (rej the negative answer).
(d) Paraffin sold = 500/x - 3 = 52 - 3 = 49.

👍 0 👎 0