HCF and compound expressions

Hi, can someone help please. I would like to know (in detail plz) how to approach the typical problems that follows, as I am not quite sure how to go about starting to solve this. I really need to know an established method about solving things like this.

question :-Find the HCF of
24x^4-2x^3-60x^2-32x and 18x^4-6x^3-39x^2-18x


(note ^ = to the power of)


thanks for any help
Gaz

First one:
24x⁴ - 2x³ - 60x² - 32x
Simple, the highest common factor must be 2x because every number is even, and 2 is the smallest co-efficient that evenly divides into every other coefficient (if it were four, then the highest common factor would be 4x for the same reason), and every component is a multiple of x:
2x × (12x³ - x² - 30x - 16)

Second one:
18x⁴ - 6x³ - 39x² - 18x
Every co-efficient is a multiple of three. Had 39x² been 36x², every co-efficient would have been a multiple of 6. The highest common factor, therefore is 3x, because 3 is the highest common factor of all coefficients, and every component is a multiple of x:
3x × (6x³ - 2x² - 13x - 6)

Have you heard of Euclid's algorithm, to find the greatest common divisor? (GCD = HCF)
The same algorithm works with polynomials too.
Basically, GCD( polynomial A, polynomial B ) = GCD( polynomial B, polynomial remainder when A is divided by B).
So, so long as you known how to divide polynomials (and therefor find the remainder) you keep doing this until te remainder is zero, then whatever you were dividing by is the answer :)

sorry Dream, it's my fault that I didn't make it clear enough, I actually meant the HCF of both equations taken together, the HCF of both of them. Sorry i did not make that clear first time around. The idea being to evaluate the HCF of both equations taken at the same time.

Quote:

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Originally Posted by tsd_man

sorry Dream, it's my fault that I didn't make it clear enough, I actually meant the HCF of both equations taken together, the HCF of both of them. Sorry i did not make that clear first time around. The idea being to evaluate the HCF of both equations taken at the same time.

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HAHAHA man I'm not terribly thoughtful at times am I... hehehe. Sorry, must have felt like I was treating you like a high schooler...

Here's a thought... Perhaps the answer to your question is then what's the HCF of the HCFs? The HCF of the first eqn is 2x and that of the second is 3x, so then the common factor between the two answers is x.

No, that's not it. You have to use polynomial division.
If you can factorise each expression that makes it trivial too...just write both as the product of polynomials, and see which factors are in common.