Reducing Algebraic Fractions

The reason that so much effort is put into factoring algebraic expressions is that when we are presented with algebraic fractions, the fraction can only be reduced (made simpler) by cancelling like factors. If a fraction has more than 1 term in the expression of the numerator or denominator then that expression must be factored. It must be factored to the point where the expression is the product of factors (1 term) instead of sum and/or difference of terms (more than 1 term). 3 terms 18x3y2 – 15x2y3 + 3xy4 18x3y2 – 15x2y3 + 3xy4 = 3xy2(6x2 – 5xy + y2) = 3xy2(6x2 – 2xy - 3xy + y2) = 3xy2( 2x(3x – y) – y(3x - y) ) = 3xy2(3x – y) (2x – y) = 3•x•y•y (3x – y) (2x – y) 6 factors but 1 term The original expression containing 3 different terms has been transformed to an expression having the same value but in the form of 6 factors. When expressions are joined by multiplication and/or division they are 1 term.

18x3y2 – 15x2y3 + 3xy4 = 3•x•y•y (3x – y) (2x – y) The expression on the left side is not in a form that allows us to identify its factors. This means that in a fraction we would not be able to determine how to simplify the fraction through reduction. When the expression is transformed by factoring like it is on the right, we are then able to identify like factors. Identical factors that appear in the numerator and denominator can be cancelled thereby reducing the fraction. When we are faced with algebraic fractions where the numerator and denominator are monomials (1 term) to start with, there is no need to factor. The factors are obvious. Like factors in the numerator and denominator can be identified and cancelled. Both the numerator and denominator have a factor of 3 and they both have a factor of y. These factors can be cancelled. Because all of the factors of the numerator have been cancelled, we write ‘1’ because there is always an invisible factor of 1 present. The denominator is written with its remaining factors – 3 and y.

The algebraic fraction presented here is not in a form that allows to identify and cancel factors. The numerator, 5m + 10, is a binomial and the denominator, m2 + m – 2, is a trinomial. We must transform the numerator and the denominator to a form that will allow us to identify the factors and thereby cancel like factors. In other words factor the numerator and denominator. We can factor the numerator by removing a common factor of 5. We can factor the denominator by using the short cut trinomial method. Factors of -2 SUM = +1 -1 +2

We can replace the expressions of the numerator and the denominator with the factored forms of each expression. Both the numerator and the denominator have 2 factors. One of the factors from the numerator is identical to one from the denominator. This factor can be cancelled.

If more space is required to follow the steps required to factor the numerator and/or denominator, it can be carried out separate from the main body of work or on scrap paper. Factors of +5 SUM = -6 -1 -5 Both the numerator and the denominator have 2 factors. One of the factors from the numerator is identical to one from the denominator. This factor can be cancelled.

Factors of +1 SUM = -2 -1 -1 Both the numerator and the denominator have 2 factors. None of the factors from the numerator is identical to any of those from the denominator. However, factors (r - s) and (s - r) look similar but they are different. They can be made the same by removing a factor of -1 from (s – r).

Factors of -2 SUM = +1 -1 +2 Factors of -3 SUM = -2 +1 -3 Both the numerator and the denominator have 2 factors. None of the factors from the numerator is identical to any of those from the denominator. This means that no factors can be cancelled. Factors (a - b) and (a + b) look a lot alike but they are different and cannot be made the same through any algebraic means.

COMMON MISTAKES b2 is an identical term in the numerator and denominator. However, it cannot be cancelled. The numerator is NOT a 1-term expression it is a 2-term expression. b + a = a + b Because these terms are identical and can be viewed as 1 term, they can be cancelled. b - a ≠ a - b Because these terms are not identical, they can not be cancelled to get 1. b - a = -1(a – b) Because these terms can be made identical by removing a common factor of -1 from (b – a), they can be cancelled to get -1.