Questions about Polynomials!!!

[匿名]

2011-02-08 10:08 pm

1.When f(x)=x^2+ax+b is divided by x-1,the remainder is -6.x+2 is a factor of f(x).

a)Find the values of a and b.
b)Find the remainder when f(x) is divided by x+3.

2.It is known that f(x) is the sum of two parts,one part varies directly as x^3 and the other part varies directly as x^2.Assume that f(-1)=1 and f(2)=16.

a)Find f(x)
b)Let g(x)=f(x)-5x-6.
i)Prove that x+3 is a factor of g(x).
ii)Factorize g(x)

THX A LOT!!!

回答 (1)

STEVIE-G™

2011-02-08 10:19 pm

✔ 最佳答案

1a)
When f(x)=x^2+ax+b is divided by x-1,the remainder is -6
=> f(1)=-6
=> (1)+a+b=-6
=> a+b=-7...(1)

x+2 is a factor of f(x)
=> f(-2)=0
=> (-2)^2+(-2)a+b=0
=> -2a+b=-4...(2)

(1)-(2): 3a=-3 => a=-1 and b=-7-(-1)=-6

b)
The remainder
=f(-3)
=(-3)^2-(-3)+(-6)
=6

2a)
f(x)=ax^3+bx^2...(*) [where a and b are constants]

Sub f(-1)=1 into (*):
1=a(-1)^3+b(-1)^2
b-a=1...(1)

Sub f(2)=16 into (*):
16=a(2)^3+b(2)^2
8a+4b=16
2a+b=4...(2)

Sub (1) into (2):
2a+(a+1)=4
3a=3
a=1 and b=2

So, f(x)=x^3+2x^2

bi)
g(x)=f(x)-5x-6=x^3+2x^2-5x-6
g(-3)=(-3)^3+2(-3)^2-5(-3)-6=0

So, x+3 is a factor of g(x)

ii)
g(x)
=x^3+2x^2-5x-6
=x^3+3x^2-(x^2+5x+6)
=x^2(x+3)-(x+2)(x+3)
=(x+3)(x^2-x-2)
=(x+3)(x-2)(x+1)

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