Simplifying Algebraic ExpressionsDate: 03/23/2002 at 19:01:45 From: Heidi Callahan Subject: Simplifying Algebraic Expressions I am looking for guidelines to follow when simplifying algebraic expressions (for simple expressions and more complex expressions). Are there certain rules to follow? Date: 03/27/2002 at 09:12:51 From: Doctor Ian Subject: Re: Simplifying Algebraic Expressions Hi Heidi, In some sense, your algebra textbook is a collection of guidelines for simplifying expressions. There's not a lot that I'd be able to add to what's already there. In my experience, the most useful 'rules' are (1) the distributive property, and (2) factoring into binomials. The former allows you to gather together like terms that are originally separated in an expression. The latter allows you to cancel identical binomials. For example, suppose we start with a mess like this: x(xy - 4x) - x(y + 4) - 3(2y + 8) -------------------------------- xy^2 - 16x - 3y^2 + 48 Applying the distributive property allows us to break everything into monomials. Once we've done that, we can move like monomials together, and use the distributive property again to factor out shared terms. The resulting polynomials are candidates for factoring into polynomials: (x^2y - 4x2) - (xy + 4x) - (6y + 24) Distribute multiplications ------------------------------------ xy^2 - 16x - 3y^2 + 48 x^2y - 4x2 - xy - 4x - 6y - 24 Eliminate parentheses ------------------------------ xy^2 - 16x - 3y^2 + 48 x^2y - xy - 6y - 4x2 - 4x - 24 Move like terms together ------------------------------ xy^2 - 16x - 3y^2 + 48 y(x^2 - x - 6) - 4(x2 - x - 6) Factor ------------------------------ xy^2 - 16x - 3y^2 + 48 y(x+2)(x-3) - 4(x+2)(x-3) Factor ------------------------------ xy^2 - 16x - 3y^2 + 48 (y-4)(x+2)(x-3) Undistribute multiplications ------------------------------ xy^2 - 16x - 3y^2 + 48 (y-4)(x+2)(x-3) ------------------------------ x(y^2 - 16) - 3(y^2 - 16) Factor (y-4)(x+2)(x-3) ------------------------------ (x-3)(y^2 - 16) Undistribute multiplications (y-4)(x+2)(x-3) ------------------------------ (x-3)(y+4)(y-4) Factor (x+2) ------------------------------ Cancel (y+4) All of which is to say, armed with only these two techniques, you can go a long, long way. Which would explain why so much emphasis is placed on them! I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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