How to Add & Subtract Rational Expressions With Unlike Denominators
Rational expressions can stump even the best math students. In reality, however, they function similarly to all other fractions. The only difference is that rational expressions use polynomials as either the denominators or numerators or both. Just like you do with other fractions, add and subtract rational expressions with unlike common denominators by first creating a common denominator.
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Instructions
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Multiply each rational expression by a fraction equal to 1, where the denominator and numerator are both equal to the denominator of the other rational expression. For example, to add 1 / 4y and 1 / (2y + 1), multiply 1 / 4y by (2y + 1) / (2y + 1) to get (2y + 1) / (8y^2 + 4y). Multiply 4y / 4y by 1 / (2y + 1) to get 4y / (8y^2 + 4y).
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Add the numerators together; place the sum over the shared denominator of the two rational expressions. For example, add the numerators from (2y + 1) / (8y^2 + 4y) and (4y) / (8y^2 + 4y) to get (6y + 1) / (8y^2 + 4y). If you want to subtract one number from another, multiply the entire subtracted numerator by -1, then add the two numerators together. Adding a negative number gets the same result as subtracting a positive number with the same absolute value (distance from zero).
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Reduce the rational expression to its lowest terms. For example, the reduction of (5y + 5) / (y^2 + 4y + 3) equals [5 (y + 1)] / [(y + 1)(y + 3)].
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Cross out the numerator that matches a denominator. Cross out the matching denominator as well. For example, in 5(y + 1) / (y + 1)(y + 3), cross out the (y + 1) expressions in both the numerator and denominator.
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Rewrite your expression in its lowest terms. By crossing out the (y + 1) expressions in both the numerator and denominator of [5(y + 1)] / [(y + 1)(y + 3)], you get 5 / (y + 3).
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Tips & Warnings
Some people find it easier to reduce the denominators into factors before beginning. In this way, you can cancel out any matching numerators and denominators at the start. Although unnecessary, this makes later simplification easier.
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