Solving Algebraic Expression with Limits

[krbs]

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

 

[mfb]

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.

 

[hilbert2]

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.

 

[krbs]

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

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?

 

[mfb]

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.

 

[krbs]

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!