Solving Algebraic Expression with Limits

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In summary, the problem involves simplifying the expression (x3-6x2+12x-8)/(x2-4x+4) to (x-2)3/(x-2)2. This can be done by finding the roots of the denominator and using polynomial long division to divide out the common factor (x-2). Alternatively, with practice, one can directly see or guess that the numerator is (x-2)3 and skip the steps of finding the roots and using polynomial long division.
  • #1
krbs
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Hi guys, there's a sample problem in m textbook where they simplify an expression from x3-6x2+12x-8/x2-4x+4 to (x-2)3/(x-2)2. Can you explain how they solved this? For reference, I'm learning about limits
 
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I moved the thread to our homework section as the problem is homework-like.

I guess you mean (x3-6x2+12x-8)/(x2-4x+4). For the denominator, you should be able to find roots, once you know where the denominator gets zero you can also write it as product (here: (x-2)(x-2)). For the numerator, guess a root, then take it out as factor and compute the other factor, then do the same as for the denominator.
 
  • #3
Knowledge of the fact that one root of a cubic polynomial with integer coefficients is a divisor of the constant term (here 8) will help in guessing.
 
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  • #4
mfb said:
I moved the thread to our homework section as the problem is homework-like.

I guess you mean (x3-6x2+12x-8)/(x2-4x+4). For the denominator, you should be able to find roots, once you know where the denominator gets zero you can also write it as product (here: (x-2)(x-2)). For the numerator, guess a root, then take it out as factor and compute the other factor, then do the same as for the denominator.

Uh, yeah, I missed the big notice at the top... Had a couple glasses of wine :oops::H

Ok, so I guessed root 2 (lol). Would I then just divide out factor (x-2), like polynomial long division, or is there an easier way I'm overlooking?
 
  • #5
Polynomial long division is the right approach - unless you directly see or guess that the numerator is (x-2)3 (possible with practice), then you can skip those steps.
 
  • #6
mfb said:
Polynomial long division is the right approach - unless you directly see or guess that the numerator is (x-2)3 (possible with practice), then you can skip those steps.

Ok thank you!
 

FAQ: Solving Algebraic Expression with Limits

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