Q&A for How to Simplify Algebraic Fractions

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  • Question
    Why does it work exactly like this all the time?
    Donagan
    Top Answerer
    Because the properties of and operations with numbers never change.
  • Question
    When factoring algebraic fractions, how do I know which one will have a plus or minus sign? For example, 12x^2 + 43x+ 35 / 6x^2 +5x - 21?
    Donagan
    Top Answerer
    This is known as "solving by inspection." In other words, you just look at ("inspect") the expression you're factoring, and choose plus and minus signs so that the factors, when multiplied together, result in the original expression. In the example cited, 12x² + 43x +35 factors into (3x + 7) and (4x + 5). 6x² + 5x - 21 factors into (3x + 7) and (2x - 3). In each case the signs were chosen so that multiplying the factors together would result in the original expression. In this case there is a factor of (3x + 7) in both the numerator and denominator, so that they cancel each other. The final, simplified expression is (4x + 5) / (2x - 3).
  • Question
    What if there's a question like this: (1–y²) / (–y² – y + 2)?
    Donagan
    Top Answerer
    First simplify a little by factoring (-1) from both numerator and denominator: -(y²-1) / -(y²+y-2). The (-1)'s will cancel each other, and you're left with (y²-1) / (y²+y-2). Factor numerator and denominator: [(y+1)(y-1)] / [(y+2)(y-1)]. The two (y-1)'s cancel each other, and the final fraction is (y+1) / (y+2).
  • Question
    How to simplify 17y/9?
    Community Answer
    That term is already in its simplest form.
  • Question
    Factorize (a - b)(a - b).
    Community Answer
    That expression is already factorized (or factored). In other words, it cannot be factorized (factored) any further. It can also be written as (a - b)².
  • Question
    How can you simplify 8y - 5 - 3y + 2?
    Community Answer
    Combine like terms: (8y - 3y) + (-5 + 2) = (5y) + (-3) = 5y - 3.
  • Question
    What is x² - y² divided by 3x + 3y?
    Community Answer
    (x² - y²) / 3x + 3y = [(x - y)(x + y)] / [3(x + y)]. The (x + y) in the numerator is canceled by the (x + y) in the denominator, and we're left with (x - y) / 3.
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