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Simplification Of Fractions
Textbooks
Mathematics Grade 10
Algebraic Expressions
Simplification Of Fractions
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1.8 Simplification of fractions (EMAQ)
We have studied procedures for working with fractions in earlier grades.
ab×cd=acbd(b≠0;d≠0)
ab+cb=a+cb(b≠0)
ab÷cd=ab×dc=adbc(b≠0;c≠0;d≠0)
Note: dividing by a fraction is the same as multiplying by the reciprocal of the fraction.
In some cases of simplifying an algebraic expression, the expression will be a fraction. For example,
x2+3xx+3
has a quadratic binomial in the numerator and a linear binomial in the denominator. We have to apply the different factorisation methods in order to factorise the numerator and the denominator before we can simplify the expression.
x2+3xx+3=x(x+3)x+3=x(x≠−3)
If x=−3 then the denominator, x+3=0and the fraction is undefined.
This video shows some examples of simplifying fractions.
Video: 2DNV
WORKED EXAMPLE 18: SIMPLIFYING FRACTIONS
Simplify:ax−b+x−abax2−abx,(x≠0;x≠b)
Use grouping to factorise the numerator and take out the common factor ax in the denominator
(ax−ab)+(x−b)ax2−abx=a(x−b)+(x−b)ax(x−b)
Take out common factor (x−b) in the numerator
=(x−b)(a+1)ax(x−b)
Cancel the common factor in the numerator and the denominator to give the final answer
=a+1ax
WORKED EXAMPLE 19: SIMPLIFYING FRACTIONS
Simplify:x2−x−2x2−4÷x2+xx2+2x,(x≠0;x≠±2)
Factorise the numerator and denominator
=(x+1)(x−2)(x+2)(x−2)÷x(x+1)x(x+2)
Change the division sign and multiply by the reciprocal
=(x+1)(x−2)(x+2)(x−2)×x(x+2)x(x+1)
Write the final answer
=1
WORKED EXAMPLE 20: SIMPLIFYING FRACTIONS
Simplify:x−2x2−4+x2x−2−x3+x−4x2−4,(x≠±2)
Factorise the denominators
x−2(x+2)(x−2)+x2x−2−x3+x−4(x+2)(x−2)
Make all denominators the same so that we can add or subtract the fractions
The lowest common denominator is (x−2)(x+2).
x−2(x+2)(x−2)+(x2)(x+2)(x+2)(x−2)−x3+x−4(x+2)(x−2)
Write as one fraction
x−2+(x2)(x+2)−(x3+x−4)(x+2)(x−2)
Simplify
x−2+x3+2x2−x3−x+4