Simplifying Polynomial Fractions

A rational expression is same as a fraction in which both the numerator or denominator are polynomials. Similarly polynomial fraction is in the form of ratio of two polynomials like where divisible of zero is not allowed, like . The same rule of a numerical fraction is applied here as of polynomial fraction, the difference is that both the numerator and denominator are polynomials. Various operations can be performed same as we do in simple arithmetic such as add, divide, multiply and subtract.

Polynomial fraction is an expression of a polynomial divided by another polynomial. Let P(x) and Q(x), where Q(x) cannot be zero.

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Polynomial fraction can be simplified with the polynomial present in the numerator or denominator by facotrising and reducing them to the lowest terms. As an example, the fraction  will be reduced to the lowest term as . Similarly, the fraction  will be simplified to  by factoring both the numerator and the denominator and reducing to lowest terms.

Polynomials within the numerator or the denominator

Step 1. To factorize polynomial within the numerator or the denominator, first factor the polynomial within the numerator or the denominator.

Step 2. Then simplify the fraction to lowest terms by canceling out any common monomials or polynomials that exist in both the numerator and denominator, if possible.

Step 3.  Finally, rewrite the terms by multiplying any monomials or polynomials that remain within the numerator or denominator. 

Example 1: Simplify the fraction 

Solution: Step 1. Factorize the polynomial completely in the denominator.

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Step 2. Reduce the fraction to lowest terms by canceling out any common monomials or polynomials that exist in both the numerator and denominator.

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Step 3. Rewrite the expression in the denominator by using distributive property.

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Example 2: Simplify the fraction 

Solution: Step 1. Factorize the polynomial completely in the denominator.

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Step 2. Reduce the fraction to lowest terms by canceling out any common monomials or polynomials that exist in both the numerator and denominator.

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Polynomials within the numerator and denominator

Step 1. To factorize polynomial within the numerator and also the denominator, factor the polynomial completely within the numerator and also the denominator.

Step 2. Then simplify the fraction to lowest terms by canceling out any common monomials or polynomials that exist in each of the numerator and denominator.

Step 3. Finally, multiply any remaining factors within the numerator or denominator.

Example 1: Simplify the fraction 

Solution: Step 1 Factorize the polynomial completly in the numerator and the denominator.

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Step 2 Reduce the fraction to lowest terms by canceling out any common monomials or polynomials that exist in both the numerator and denominator.

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Step 3. Rewrite the expression in the numerator by using distributive property

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Example 2: Simplify the fraction 

Solution: Step 1 Factorize the polynomial completely in the numerator and the denominator.

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Step 2 Reduce the fraction to lowest terms by canceling out any common monomials or polynomials that exist in both the numerator and denominator.

= 

Step 3. Rewrite the expression in the denominator by using distributive property

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Exercise

Simplify the following polynomial:

1. 
2. 
3. 
4. 
5. 
6.