If the numerator and denominator of an algebraic fraction have no common factor other than 1, it is said to be in the lowest form.
The reduced form of an algebraic fraction means there is no common factor between the numerator and denominator of the given algebraic fractions. That means, if there is any common factor present in the numerator and denominator then by keeping the value of the algebraic fraction unchanged, the common factor is freed by mathematical method and the algebraic fraction will be reduced to its lowest form.
When we reduce an algebraic fraction to its lowest term we need to remember if the ‘numerator’ and ‘denominator’ of the fractions are ‘multiplied’ or ‘divided’ by the same quantity, then the value of the fraction remains unchanged.
To reduce algebraic fractions to its lowest term, we need to follow the following steps:
Step I: take the factorization of polynomial in the numerator and denominator.
Step II: then cancel out the common factors in the numerator and denominator.
Step III: reduce the given algebraic fraction to the lowest term.
Note: The H.C.F. of numerator
and denominator is 1.
For example:
1. In the numerator ma and the denominator mb of \(\frac{ma}{mb}\), is the common factor, so the algebraic fraction \(\frac{ma}{mb}\) is not in its lowest terms. Now, divide both the numerator and denominator by the common factor ‘m’ then we get \(\frac{ma ÷ m}{mb ÷ m}\) = \(\frac{a}{b}\) there is no common factor, so \(\frac{a}{b}\) is the algebraic fraction which is in reduced form.
2. \(\frac{x^{3} + 9x^{2} + 20x}{x^{2} + 2x  15}\)
We see that the numerator and denominator of the given algebraic fraction is polynomial, which can be factorized.
= \(\frac{x(x^{2} + 9x + 20)}{x^{2} + 5x  3x  15}\)
= \(\frac{x(x^{2} + 5x + 4x + 20)}{x^{2} + 5x  3x  15}\)
= \(\frac{x[x(x + 5) + 4(x + 5)]}{x(x + 5) – 3(x + 5)}\)
= \(\frac{x(x + 5)(x + 4)}{(x + 5) (x – 3)}\)
We observed that in the numerator and denominator of the algebraic fraction, (x + 5) is the common factor and there is no other common factor. Now, when the numerator and denominator of the algebraic fraction is divided by this common factor or their H.C.F. the algebraic fraction becomes,
= \(\frac{\frac{x{(x + 5) (x + 4)}}{(x + 5)}}{\frac{(x + 5) (x  3)}{(x + 5)}}\)
= \(\frac{x(x + 4)}{(x – 3)}\), which is the lowest form of the given algebraic fraction.
8th Grade Math Practice
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