f4 polynomial questions

[匿名]

2008-08-22 10:51 pm

1) Write a polynomial of degree 3 with 2x +3 as one of its factors.

2) It is givedn that f(x) = x^3 + mx^2 +nx +5, where m and n are integers. If f(x) is divisible by x-1, list two sets of possible values of m and n.

3) if x +2 is a factor of x^3 + ax^2 -x -b, where a and b are integers, list two sets of possible values of a and b.

pls. solve the question step by step and explain if possible.
thanks a lot!

回答 (1)

momo

2008-08-22 11:28 pm

✔ 最佳答案

(1)
polynomial
= (2x+3)^3 = (2x)^3 + (3C1)(2x)^2(3) + (3C2)(2x)(3)^2 + (3)^3
= 8x^3 + 36x^2 + 54x + 27
(2x+3) is one of the factor of the polynomial
(2) f(x) = x^3 +mx^2 + nx + 5
f(1) = 1^3 + m(1)^2 + n(1) + 5 = 0
m+ n = -6
Choose n = 1, m = -7
Choose m = 1, n = -7
when n = 1 and m = -7, f(x) = x^3 -7x^2 + x + 5
when n = -7 and m = 1, f(x) = x^3 + x - 7x + 5
(3)
Let f(x) = x^3 + ax^2 - x - b
f(-2) = (-2)^3 + a(-2)^2 -(-2) - b = 0
4a - b = 6
Choose b = 2, a = 2
Choose a = 1, b = -2
When a = 2, b = 2, f(x) = x^3 + 2x^2 - x - 2
when a = 1, b = -2, f(x) = x^3 + x^2 - x + 2

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