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When presented with the following expression :

$$\frac{1}{(x+1)(x+2)}-\frac{2}{x+1}+\frac{3}{x+2}$$

I have used an online calculator to show me the LCD, which is :

$$(x+2)(x+1)$$

Although I can work out the LCD for different algebraic expressions, I am unsure of the methodology behind solving this one. I realise it must be basic but it just has not clicked yet.

If anybody could explain how this is done I would be very grateful.

Aztec warrior
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1 Answers1

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LCD - lowest common denominator. Looking at the given denominators, (x+2)(x+1), (x+1) and (x+2) Their lowest common multiple (LCM) is the LCD of the three fractions, in this case (x+2)(x+1)

For an exact approach, we can use Euclidean algorithm to get the Greatest Common Divisor (GCD). The LCM is basically the product of all three numbers divided by the GCD

In this case, by the Euclidean algorithm, the GCD is (x+2)(x+1). The product of the three numbers is (x+2)(x+1)(x+2)(x+1). Dividing this by the GCD we get simply (x+2)(x+1).

drewyu
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  • how do you know that the lowest common multiple is (x+2)(x+1), what did you do to work this out ? – Aztec warrior Oct 16 '16 at 17:05
  • For an exact approach, we can use Euclidean algorithm to get the Greatest Common Divisor (GCD). The LCM is basically the product of all three numbers divided by the GCD – drewyu Oct 16 '16 at 17:07
  • In this case, by the Euclidean algorithm, the GCD is (x+2)(x+1). The product of the three numbers is (x+2)(x+1)(x+2)(x+1). Dividing this by the GCD we get simply (x+2)(x+1). Hope this helped – drewyu Oct 16 '16 at 17:11
  • Thank you this is clear to me now :) – Aztec warrior Oct 16 '16 at 17:59