Factoring Cubic Polynomials/Function

  • #1

Homework Statement


Doing my vector mechanics dynamics homework and I cannot believe I am stuck on this part.

trying to factor

t^3 - 6t^2-36t - 40 = 0

Homework Equations




The Attempt at a Solution


I honestly do not remember where to begin. I remember there is a formula for perfect cubes, but do not remember what is needed for this situation. Any tips is greatly appreciated.
 

Answers and Replies

  • #2
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There is a formula but it's nasty and you could always just try Newton-Raphson method for the first factor then long division?
 
  • #3
vela
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You can find all the factors of 40 to try guess how the polynomial factors. Generally, though, I find plotting the function is the quickest way of finding its roots, at least for homework problems.
 
  • #4
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Easy way is to find one root and divide the cubic by it.

i.e. in this situation you can see -2 is a root, so divide the polynomial by t+2, then factor the result to get the other roots.
 
  • #5
Mentallic
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If the problem were changed to +36t, then you would be able to factorize by using the perfect cube.
You would end up having [itex](t-6)^3+176=0[/itex] and then you could use the sum of two cubes formula to factorize further (or even just solve the equation for t directly).

But this is not the case :smile:
 
  • #6
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Well the simplest way is to find factors of -40. Replace each in the original expression one by one and when the result is zero: Bingo! You've found a root. In this case 10 is one root. next, divide the original expression by x-r (r = root found by trial and error) to get a quadratic equation. solve it and you've got your roots:smile:
 

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