How to Simplify Algebra 2
In order to more quickly solve an algebraic equation, and sometimes to solve it at all, you must use a simplified version of the equation. Unfortunately, real life situations and math books do not usually supply equations in this simplified form. Therefore, much of the material in Algebra 2 classes is dedicated to simplifying the expressions that form equations. Furthermore, the functions in Algebra 2 revolve around more varied and complex forms than those of Algebra 1 and call for several very different techniques.
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Instructions
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To simplify linear equations and polynomials, distribute, FOIL, and combine like terms when necessary. When multiplying variables, use the properties of exponents. When the variable (it has to be the same variable) is being multiplied, add the exponents. If you are raising a variable to a second power, multiply the exponents. For example, X^5 x X^3 simplifies to X^8 and (x^5)^3 simplifies to X^15.
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To simplify a radical expression (one that involves roots), create a factor tree for any numbers inside the radicals. Then, take out sets of factors that correspond to the number of the root. For example, if it is a cube root, you would look for any tripled factors and write one copy for each triple outside the radical. For variables, divide the exponent of the variable inside by the number of the root - the answer becomes the exponent of the variable when it is written on the outside, and the remainder (if there is one) becomes the exponent of the variable left on the inside. Then, multiply any numbers on the outside of the various radicals and combine any multiplied variables using exponent properties. Finally, add together or subtract any radicals that have the exact same numbers/variables/exponents on the inside as though they are like terms.
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To simplify rational expressions that look like algebraic fractions, cancel any common factors that the numerator and denominator share. When exponents are involved, remember that the exponent from the denominator is subtracted from the exponent in the numerator of the same variable to create a new exponent in the numerator for that variable (X^5/X^2 = X^3). Switch sides of the division bar for, and make positive, any variable that has a negative exponent (4X^-3/y^2 would become 4/X^3Y^2). Factor any polynomials in the numerator or denominator so that you can look for whole factors that cancel.
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To simplify a complex fraction (division within division), find the Least Common Denominator for every term involved in the expression. Then, change each numerator as necessary, so that each term is over the LCD. Next, eliminate all the LCDs so that you are left with only the new numerators. Finally, distribute, FOIL, and combine any like terms.
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To simplify a logarithmic expression, compress multiple logarithms into one using the properties of logarithms. Any "logs" that are being added are then multiplied together inside the single logarithmic function. Any that are being subtracted, are divided in the compressed function. Lastly, any coefficients in front of a log function become exponents in the final log. For example 3log(X) - 2log(Y) + log(3) would become: log(3X^3/Y^2). Also, any log of the same number as the base simplifies to the number one, and any log of the number one simplifies to zero.
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Tips & Warnings
When dealing with rational expression, do not try to cancel things that are being added or subtracted. Only things that are being multiplied can cancel.
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