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Simplifying algebraic fractions

Simplifying algebraic fractions

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Explainer Video

Tutor: Toby

Summary

Simplifying algebraic fractions

In a nutshell

Algebraic fractions are fractions that can be represented as the ratio of two polynomials. Prior to operating with them, it is useful to factorise the numerator and denominator in order to simplify the process. 



Cancelling factors in algebraic fractions

Procedure

1.
Factorise the numerator and/or denominator where possible. 
2.
Identify common factors and cancel them. 
3.
Simplify where possible.


Example 1

Simplify this fraction as far as possible x2+9x+20x+4\cfrac{x^2 +9 x + 20}{x+4}x+4x2+9x+20​​


First you factorise numerator and denominator. The denominator is already factorised, so you only need to focus on the numerator. 


The numerator x2+9x+20x^2 + 9x +20x2+9x+20 factorises into (x+4)(x+5)(x+4)(x+5)(x+4)(x+5). Therefore, you have 

x2+9x+20x+4=(x+4)(x+5)x+4\cfrac{x^2 +9 x + 20}{x+4} = \cfrac{(x+4)(x+5)}{x+4}x+4x2+9x+20​=x+4(x+4)(x+5)​​​


Now cancel the common factors. 

​(x+4)(x+5)x+4=(x+4)(x+5)x+4 \cfrac{(x+4)(x+5)}{x+4} = \cfrac{\cancel{(x+4)}(x+5)}{\cancel{x+4}} x+4(x+4)(x+5)​=x+4​(x+4)​(x+5)​​​


In conclusion, x2+9x+20x+4=x+5‾\underline{\cfrac{x^2 +9 x + 20}{x+4} = x+5}x+4x2+9x+20​=x+5​​​


Example 2

Simplify this fraction 8x3−4x2−6x2x\cfrac{8x^3 - 4x^2 -6x}{2x}2x8x3−4x2−6x​​


You have that xxx is a common factor, so it can be eliminated. This is also true of 222:

8x3−4x2−6x2x=x(8x2−4x−6)2x=2(4x2−2x−3)2=4x2−2x−3\cfrac{8x^3 - 4x^2 -6x}{2x} = \cfrac{\cancel x (8x^2 - 4x- 6)}{2\cancel x}= \cfrac{\cancel 2 (4x^2 -2x -3)}{\cancel 2} = 4x^2 -2x -32x8x3−4x2−6x​=2x​x​(8x2−4x−6)​=2​2​(4x2−2x−3)​=4x2−2x−3​​


As 4x2−2x−3 4x^2 -2x -34x2−2x−3 cannot be factorised into an expression (Ax+B)(Cx+D)(Ax+B)(Cx+D)(Ax+B)(Cx+D) where AAA, BBB, CCC and DDD are integers, this is the fully factorised expression.

​​​


Therefore, 8x3−4x2−6x2x=4x2−2x−3‾\underline{\cfrac{8x^3 - 4x^2 -6x}{2x} = 4x^2 -2x -3}2x8x3−4x2−6x​=4x2−2x−3​



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Algebraic notation

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Algebraic fractions - Higher

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FAQs - Frequently Asked Questions

What are the steps to follow when cancelling factors in algebraic fractions?

First, you have to factorise denominator and numerator where possible. Next you have to identify common factors and cancel them. Finally, you have to simplify where possible your expression.

What is the first step when cancelling factors in algebraic fractions?

The first step is to factorise numerator and denominator where possible.

What are algebraic fractions?

Algebraic fractions are fractions that can be represented as the division between two polynomials.

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