Simplifying Rational Expressions
Simplifying rational expressions involves transforming complex mathematical ratios into simpler, more manageable forms. A rational expression is typically represented as a fraction where both the numerator and denominator are polynomials or integers. The process allows for the reduction of expressions by identifying and eliminating common factors from the numerator and denominator. For instance, a fraction like 27/18 can be simplified by dividing both the numerator and denominator by their greatest common factor, resulting in a simplified form of 3/2. This principle applies equally to algebraic expressions, where factors common to the numerator and denominator can be canceled out.
To successfully simplify algebraic rational expressions, one may need to factor both the numerator and denominator first. For example, an expression like [x² - 9] / [x² + 7x + 12] requires initial factoring to yield a simplified form. Simplifying rational expressions does not alter their original value, making this skill essential for making calculations more efficient and accessible. Overall, understanding how to simplify these expressions is a valuable mathematical skill that aids in various applications across different contexts.
Simplifying Rational Expressions
Simplifying rational expressions is a skill that is used to transform complex mathematical expressions into easier, more usable forms. A mathematical expression is a written statement that contains numbers, letters, and perhaps other symbols to represent a quantity, variable situation, or other mathematical idea. A mathematical expression involving a ratio is a rational expression. The skill of simplifying rational expressions is useful in transforming difficult to understand or unwieldy expressions into less complex forms.
Simplifying Fractions
A fraction, the simplest and most common type of rational expression, is a ratio between two numbers and is usually written as a/b, where a and b are integers and b ≠ 0. The number on top (a) is called the numerator, and the number on the bottom (b) is called the denominator. For example, 1/2, 33/5, and −4/7 are all fractions. Further, these fractions are all in simplified form, meaning they have no common factor in the numerator and denominator. For example, 33 and 5 have no common factor other than 1 (the factors of 33 are 1, 3, 11, and 33; the factors of 5 are 1 and 5), so the fraction 33/5 cannot be simplified.
An example of a fraction that is not in simplified form is 27/18, because 27 and 18 have common factors; specifically, 3 and 9 are numbers that divide evenly into both 18 and 27. We simplify a fraction by dividing the numerator and the denominator by the largest common factor (this is often called factoring out), which in this case would be 9. Therefore, 27 ÷ 9 = 3, and 18 ÷ 9 = 2, so the fraction 27/18 can be simplified to 3/2. The fractions 27/18 and 3/2 have the same value, even though they are written differently. The process of simplifying a fraction does not change its value.
Simplifying Algebraic Expressions
Some algebraic expressions are written as fractions. These expressions are simplified in exactly the same way that we simplify numeric fractions—by factoring out the common factors. The easiest types of algebraic rational expressions to simplify are of the following nature: x5/x3. Here we see that there are three factors of x common to the numerator and denominator, which when factored out leaves x2/1, or simply x2. A more complex rational expression is:
Here we see that there is a common factor of (x − 3) in the numerator and denominator, so we can simplify this expression to (x + 4)/(x + 7). As in the case of numeric fractions, this is accomplished by dividing the numerator and denominator by the common factor (x − 3). It is also the case that the basic value of the expression does not change through simplification.
Sometimes factoring is required before simplification can occur. For example, the numerator and the denominator of the expression [x2 – 9]/[(x2 + 7x + 12)] must first both be factored in order to simplify it. Specifically, we rewrite the expression as [(x + 3)(x − 3)]/[(x + 3)(x + 4)], then factor out (x + 3) to obtain the simplified expression (x − 3)/(x + 4).
Bibliography
Brousseau, Guy, Virginia Warfield. Teaching Fractions Through Situations: A Fundamental Experiment. New York: Springer, 2013.
Fosnot, Catherine T., and Bill Jacob. Young Mathematicians at Work: Constructing Algebra. Portsmouth, NH: Heinemann, 2010.
Kalman, Richard S. "Teaching Algebra Without algebra." Mathematics Teaching in the Middle School 13(6) (2008): 334-339.
Petit, Marjorie M. A Focus on Fractions: Bringing Research to the Classroom. New York: Routledge, 2010.
Small, Marian.Uncomplicating Fractions to Meet Common Core Standards in Math. New York: Teachers College, 2013.