Solving (b) (i) and (ii): Find Value of k and LCM of f(x) and f(x+k)

[kenny1999]

Homework Statement

given f(x) = x^3 - 6x^2 + 3x + 10

(a) Factorize f(x)
(b) if k is a positive integer such that HCF of f(x) and f(x+k) is linear.
(i) find value of k
(ii) find LCM of f(x) and f(x+k)

Homework Equations

I have no problem with (a) and f(x) = (x+1) (x^2 - 7x + 10)

I have problem with (b) (i) and (ii), I only know that HCF is linear means that there is only one factor which is linear. And that's it. I know nothing further to solve the questipn

The Attempt at a Solution

 

[eumyang]

Homework Equations

I have no problem with (a) and f(x) = (x+1) (x^2 - 7x + 10)

Do you need a complete factorization? If so, you are not finished. x² - 7x + 10 is factorable.

 

[kenny1999]

Do you need a complete factorization? If so, you are not finished. x² - 7x + 10 is factorable.

i have factored f(x) into (x+1)(x-5)(x-2)
but no idea to go further

 

[Ray Vickson]

i have factored f(x) into (x+1)(x-5)(x-2)
but no idea to go further

What are the factors of f(x+k)?

 

[eumyang]

Using the factored form of f(x), find f(x+k).

I'm assuming that HCF=GCF (greatest common factor). You are looking for a positive k such that there is a linear common factor between f(x) and f(x+k). So one of the three factors of f(x) should also be a factor of f(x+k). I used trial and error to find such a k. Find a k so that when you plug into the factored form of f(x+k), one of its factors is also a factor of f(x).


(Hope I'm not giving away too much here.)

 

[Ray Vickson]

Using the factored form of f(x), find f(x+k).

I'm assuming that HCF=GCF (greatest common factor). You are looking for a positive k such that there is a linear common factor between f(x) and f(x+k). So one of the three factors of f(x) should also be a factor of f(x+k). I used trial and error to find such a k. Find a k so that when you plug into the factored form of f(x+k), one of its factors is also a factor of f(x).


(Hope I'm not giving away too much here.)

Yes, I think you are.

 

[kenny1999]

hi how about LCM?

 

[Ray Vickson]

hi how about LCM?

Well, what about it? What work have you done on it so far?