How to Simplify Rational Expressions

Rational expressions are also known as algebraic expressions. A rational expression is the quotient of two polynomials. It is important to remember that the divisor can not be zero. A quotient is the answer to a division problem. Polynomials are expressions with the sum of powers with at least one variable multiplied by coefficients.To simplify an expression simply means to reduce to the simplest form where there is no other common factor between the numerator and the denominator except for 1 or -1.

Instructions

Difficulty: Challenging

How to Simplify Rational Expressions

1. Original problem: 9x^3-6x^2+x/9x^3-x=Factor what is in the numerator. x(9x^2-6x+1)=9x^3-6x^2+x This is done by finding the greatest common factor (GCF) and then multiplying the GCF and a second factor. The two factors should yield the original numerator as the product (the answer to a multiplication problem). In this case, the GCF is x. When we multiply x (the GCF) and 9x^2-6x+1 (the second factor), we get 9x^3-6x^2+x which is the original numerator. x(3x-1) (3x-1)=(9x^2-6x+1). We continue factoring until we can no longer factor any of the terms. The second factor (9x^2-6x+1), can be factored by multiplying (3x-1) and (3x-1) to get (9x^2-6x+1). Now, the numerator has been factored to the lowest terms which is x(3x-1) (3x-1).
2. Factor what is in the denominator. 9x^3-x=x(9x^2-1)This is done by finding the greatest common factor (GCF) and then multiplying the GCF and a second factor. The two factors should yield the original numerator as the product (the answer to a multiplication problem). In this case, the GCF is x. When we multiply x (the GCF) and 9x^2-1 (the second factor), we get 9x^3-x which is the original denominator. x(3x-1) (3x+1)=x(9x^2-1).We continue factoring until we can no longer factor any of the terms. The second factor (9x^2-1), can be factored by multiplying (3x-1) and (3x+1) to get (9x^2-1). Now, the denominator has been factored to the lowest terms which is x(3x-1) (3x+1).
3. Put the factored numerator and the factored denominator together. The factored numerator is x(3x-1) (3x-1). This will go on the top of the rational expression. The factored denominator is x(3x-1) (3x+1). This will go on the bottom of the rational expression. Therefore, the factored rational expression will look like this: x(3x-1) (3x-1)/x(3x-1) (3x+1).
4. Divide by the GCF. The GCF in this case is x(3x-1). When x(3x-1) is divided by the numerator and the denominator, we get 3x-1/3x+1.Explanation: any number divided by itself is 1 so x(3x-1)/x(3x-1) is 1 and 1 multiplied by any number or expression is equal to the number or the expression. 1 multiplied by 3x-1 is 3x-1, and 1 multiplied by 3x+1 is 3x+1.Answer: 3x-1/3x+1.