Math questions on the chapter Polynomials: Factor theorem, who knows how to answer these questions?

MandM

2011-10-12 12:26 pm

(1) If the polynomial p(x) = x^2 + ax +1 is a factor of T(x) = 2x^3 - 16x +b, find the values of a and b.

2(a)

Prove that if x^3 + mx + n is divisible by (x-k)^2, then (m/3)^3 + (n/2)^2 = 0

(b)

Prove that if x^3 + mx +n and 3x^2 + m have a common factor (x-k) then 4m^3 + 27n^2 = 0.

回答 (1)

[匿名]

2011-10-12 1:03 pm

✔ 最佳答案

1) The second factor of T must be linear of the form (2x+c) so
2x^3-16x+b=(2x+c)(x^2+ax+1)=2x^3+(c+2a)x^2+(ac+2)x+c
Compare the coefficients giving 0=c+2a, -16=ac+2, b=c
Solve these simultaneously to give a=3,b=-6 or a=-3, b=6

2a) By inspection x^3+mx+n=(x-k)^2(x+ n/k^2)=x^3+(n/k^2-2k)x^2+(k^2-2n/k)x+n
Comparing terms: n/k^2-2k=0 and k^2-2n/k=m
Find k from the first equation and sub into the second to give the answer after some bother

3) If the factor is x-k then f(k)=0 and g(k)=0
giving k^3+mk+n=0 and 3k^2+m=0
From the second equation k^3=-km/3
sub into first: -km/3+mk+n=0 this gives 2mk=-3n or k=-3n/2m
sub into 3k^2+m=0 to get result

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