Factoring High-degree Polynomials With Big Numbers

[precalckid]

Hi,

I'm struggling on this specific problem that's asked me to completely factor this polynomial:

8x⁴-92x³+342x²-513x+270

My school textbook says to use the rational root theorem and keep testing different possible factors but with a complex polynomial like this, testing out different options is a big time-waster. I'm not sure how to solve this problem - I tried doing grouping too but it didn't end up working. Also, it would be great for some advice on solving similar polynomials

Thx!

 

[Deleted member 4993]

Hi,

I'm struggling on this specific problem that's asked me to completely factor this polynomial:

8x⁴-92x³+342x²-513x+270

My school textbook says to use the rational root theorem and keep testing different possible factors but with a complex polynomial like this, testing out different options is a big time-waster. I'm not sure how to solve this problem - I tried doing grouping too but it didn't end up working. Also, it would be great for some advice on solving similar polynomials

Thx!

Use a "spread-sheet software" (e.g. Excel) to evaluate function .

 

[Cubist]

If you only have access to a simple calculator (in a test) then you are unlikely to get a problem this difficult.

However, you can always speed up a manual search by initially plugging in potential values of x that are reasonably far apart. If the polynomial value switches from +ve to -ve between two of your x values then you know that there will be (at least) one root between these two x values. So choose your next x value approximately half way between them (but still choose a value that obeys the rational root theorem). Repeat the approximate halving process until you find a root. This is a similar method to a "binary search".

EDIT: I guess you could end up "zooming in" on an irrational root - if so, then choose another location to search.

 

[Cubist]

If you only have access to a simple calculator (in a test) then you are unlikely to get a problem this difficult.

However, you can always speed up a manual search by initially plugging in potential values of x that are reasonably far apart. If the polynomial value switches from +ve to -ve between two of your x values then you know that there will be (at least) one root between these two x values. So choose your next x value approximately half way between them (but still choose a value that obeys the rational root theorem). Repeat the approximate halving process until you find a root. This is a similar method to a "binary search".

EDIT: I guess you could end up "zooming in" on an irrational root - if so, then choose another location to search.

I just tried my suggested method with your polynomial and I found it very hard to find the fractions that were close to the value that I wanted. (Mental arithmetic isn't a stong point of mine.) But I did actually "luck out" and I found a root very quickly. Nonetheless I'd adapt my advise:-

Keep track of all the fractions that have been evaluated, but initially try to concentrate efforts in "rough values of x" where the magnitude of the polynomial is lower.

And after finding one root, then divide to find a simpler polynomial!

 

[Steven G]

You have to use the rational root test, possible in conjuction with Descartes rule of signs. Look at the graph of the function to get an idea of where the roots are and then pick accordingly from the list of possible rational roots.

 

[pka]

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