Algebraic fractions are nothing but fractions where its numerator and denominator are algebraic polynomial expressions. These fractions are subject to the laws of arithmetic expressions. Reducing algebraic fractions to their lowest term means the fraction has no common factor other than 1. Get the definition, step by step process to reduce any type of algebraic fractions to its lowest term easily, few example questions and answers in the following sections.
Reduce Algebraic Fractions to its Lowest Term
Reducing algebraic fractions to its lowest term and simplifaction of algebraic fractions are one and the same. Here, we are changing the numerator, denominator of the given fraction so that it should not have a common factor between the numerator and denominator. Actually, the reduced form of the fraction is always equal to the original fraction. While we reduce an algebraic fraction to its lowest term we need to remember one important thing i.e if the numerator and denominator of the fractions are multiplied or divided by the same quantity, then the fraction value remains unchanged.
How to Reduce Algebraic Fractions to Lowest Terms?
Go through the below-mentioned steps to get help with reducing a fraction to its lowest term manually. You will find the procedure quite easy with the detailed explanation provided. Remember that the simplified algebraic fraction numerator, denominator H.C.F is always 1.
- Let us take any polynomial algebraic fraction having numerator and denominator.
- Find the factors of numerator and denominator separately.
- Cancel the common factors in both numerator and denominator.
- Reduce the algebraic fraction to the lowest term.
Solved Examples on Simplifying Algebraic Fractions
Example 1.
Simplify (x² + 5x) / (x² – 5).
Solution:
Given algebraic fraction is (x² + 5x) / (x² – 5).
We see that the numerator and denominator of the given algebraic fraction is polynomial, which can be factorized.
= [x (x + 5)] / [(x – 5) (x + 5)]
Cancel the common factor (x + 5) in the numerator, denominator.
= x / (x – 5)
Example 2.
Reduce the algebraic fraction (x² + 15x + 56) / (x² + 5x – 24) to its lowest term.
Solution:
Given algebraic fraction is (x² + 15x + 56) / (x² + 5x – 24)
Both numerator and denominator of the fraction are polynomials, which can be factorized.
= (x² + 8x + 7x + 56) / (x² + 8x – 3x – 24)
= (x(x + 8) + 7 (x + 8)) / (x(x + 8) – 3 (x + 8))
= ((x + 8) (x + 7)) / ((x + 8) (x – 3))
Cancel the common term (x + 8) in the numerator, denominator.
= (x + 7) / (x – 3)
(x² + 15x + 56) / (x² + 5x – 24) = (x + 7) / (x – 3).
Example 3.
Reduce the algebraic fraction (2x⁵ – 2x⁴ – 4x³) / (x⁴ – 1) to the lowest term.
Solution:
Given algebraic fraction is (2x⁵ – 2x⁴ – 4x³) / (x⁴ – 1)
Both numerator and denominator are polynomials, factorize them.
= (2x³ (x² – x – 2)) / ((x²)² – 1²)
= (2x³ (x² – 2x + x – 2)) / ((x² – 1) (x² + 1))
= (2x³ (x (x – 2) + 1 (x – 2)) / ((x² – 1²) (x² + 1))
= (2x³ (x + 1) (x – 2)) / ((x – 1) (x + 1) (x² + 1))
cancel the common term (x + 1) in the numerator, denominator.
= (2x³ (x – 2)) / ((x – 1) (x² + 1))
(2x⁵ – 2x⁴ – 4x³) / (x⁴ – 1) = (2x³ (x – 2)) / ((x – 1) (x² + 1))
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