Introduction. This chapter focuses on developing your skills with Algebraic Fractions

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2 Introduction This chapter focuses on developing your skills with Algebraic Fractions At its core, you must remember that sums with Algebraic Fractions follow the same rules as for numerical versions You will need to apply these alongside general Algebraic manipulation

3 You need to be able to rewrite Fractions in their simplest form One way is to find common factors, and divide the fraction by them. The factors must be common to every term. Divide by the common Factor (4) Eample Questions Divide by the common Factor (4) In the second eample, you cannot just cancel the s as they are not common to all 4 terms. If you Factorise, you can then divide by the whole Numerator, along with the equivalent part on the Denominator Factorise the Denominator Divide by the common Factor ( + ) 6 ( ) Factorise the Denominator Divide by the common Factor ( + ) A

4 You need to be able to rewrite Fractions in their simplest form One way is to find common factors, and divide the fraction by them. The factors must be common to every term. Sometimes you may have Fractions within Fractions. Find a common multiple you can multiply to remove these all together (in this case, 6) Multiply the Numerator and Denominator by 6 Factorise Divide by ( + ) Eample Questions 6 4 ( ) ( ) Multiply the Numerator and Denominator by 6 Factorise Divide by ( + ) A

5 You need to be able to rewrite Fractions in their simplest form One way is to find common factors, and divide the fraction by them. The factors must be common to every term. Sometimes you will have to Factorise both the Numerator and Denominator. Factorise the Numerator AND Denominator Eample Questions 4 ( )( ) ( )( ) Factorise the Numerator AND Denominator Divide by ( + ) Divide by ( + ) A

6 You need to be able to rewrite Fractions in their simplest form One way is to find common factors, and divide the fraction by them. The factors must be common to every term. Another Eample of a Fraction within a Fraction You will usually be told what form to leave your answer in Multiply the Numerator and Denominator by Factorise Eample Questions ( )( ) ( ) Multiply the Numerator and Denominator by Factorise Divide by ( + ) Divide by ( + ) Split the Fraction up A

7 You need to be able to multiply and divide Algebraic Fractions The rules for Algebraic versions are the same as for numerical versions When multiplying Fractions, you multiply the Numerators together, and the Denominators together It is possible to simplify a sum before you work it out. This will be vital on harder Algebraic questions a) b) c) Eample Questions 5 a c b d ac bd B

8 You need to be able to multiply and divide Algebraic Fractions The rules for Algebraic versions are the same as for numerical versions When multiplying Fractions, you multiply the Numerators together, and the Denominators together It is possible to simplify a sum before you work it out. This will be vital on harder Algebraic questions Eample Questions a c b a d) c b e) ( )( ) ( ) Factorise Multiply Numerator and Denominator B

9 You need to be able to multiply and divide Algebraic Fractions The rules for Algebraic versions are the same as for numerical versions When multiplying Fractions, you multiply the Numerators together, and the Denominators together It is possible to simplify a sum before you work it out. This will be vital on harder Algebraic questions a) Eample Questions When dividing Fractions, remember the rule, Leave, Change and Flip Leave the first Fraction, change the sign to multiply, and flip the second Fraction. B

10 You need to be able to multiply and divide Algebraic Fractions The rules for Algebraic versions are the same as for numerical versions When multiplying Fractions, you multiply the Numerators together, and the Denominators together b) Eample Questions a b a b a c c a c b It is possible to simplify a sum before you work it out. This will be vital on harder Algebraic questions When dividing Fractions, remember the rule, Leave, Change and Flip Leave the first Fraction, change the sign to multiply, and flip the second Fraction. B

11 You need to be able to multiply and divide Algebraic Fractions The rules for Algebraic versions are the same as for numerical versions When multiplying Fractions, you multiply the Numerators together, and the Denominators together It is possible to simplify a sum before you work it out. This will be vital on harder Algebraic questions When dividing Fractions, remember the rule, Leave, Change and Flip Leave the first Fraction, change the sign to multiply, and flip the second Fraction. c) Eample Questions ( 4)( 4) 4 ( ) ( 4) Leave, Change and Flip Factorise Multiply the Numerators and Denominators B

12 You need to be able to add and subtract Algebraic Fractions The rules for Algebraic versions are the same as for numerical versions When adding and subtracting fractions, they must first have the same Denominator. After that, you just add/subtract the Numerators. a) Eample Questions Multiply all by 4 Add the Numerators Multiply all by Add the Numerators C

13 You need to be able to add and subtract Algebraic Fractions The rules for Algebraic versions are the same as for numerical versions When adding and subtracting fractions, they must first have the same Denominator. After that, you just add/subtract the Numerators. b) Combine as a single Fraction Eample Questions a b a b a a b b Imagine b as a Fraction Multiply all by Combine as a single Fraction C

14 You need to be able to add and subtract Algebraic Fractions The rules for Algebraic versions are the same as for numerical versions When adding and subtracting fractions, they must first have the same Denominator. After that, you just add/subtract the Numerators. c) Factorise so you can compare Denominators Multiply by ( - ) Epand the bracket, and write as a single Fraction Simplify the Numerator Eample Questions 4 Factorise so you can compare Denominators 4 ( )( ) ( ) 4 ( ) ( ) ( )( ) 4 ( )( ) 7 ( )( ) Epand the bracket, and write as a single Fraction Simplify the Numerator C

15 You need to remember how to divide using Algebraic long division We are now going to look at some algebraic eamples.. ) Divide by ( ) So the answer is + 5, and there is no remainder This means that ( ) is a factor of the original equation Third, First, Second, Divide - 5 by by by - 5 We then subtract ( We then ) from subtract what -( we 5( started ) from with what we have left D

16 You need to remember how to divide using Algebraic long division Always include all different powers of, up to the highest that you have Divide by ( ) You must include 0 in the division So our answer is + +. This is commonly known as the quotient First, Second, Third, divide by by 4 = 0 Then, work out ( ( ) ) and and subtract from from what what you have you left started have left with D

17 You need to remember how to divide using Algebraic long division Sometimes you will have a remainder, in which case the epression you divided by is not a factor of the original equation Find the remainder when; is divided by ( 4) So the remainder is First, Second, Third, divide -4 by by = -4 = Then, work out -4( ( Then, 4) and work subtract out ( from 4) what and you subtract started have left from with what you have left D

18 You need to remember how to divide using Algebraic Long Division 9 5 = 4 5 The remainder is the numerator But, how do we deal with the remainder? 5 divides into 9 whole times The divisor is the denominator Another way to think of this sum is 9 = ( 5) = 8 The remainder is the numerator divides into 6 8 whole times The divisor is the denominator Another way to think of this sum is 6 = (8 ) + D

19 You need to remember how to divide using Algebraic Long Division We did this division earlier So the sum we have including the remainder is: Remainder Divisor ( 4) = = D

20 You need to remember how to divide using Algebraic Long Division Write ( ) ( ) in the form: A B C D Just do the division as normal 6 6 Multiply both sides by ( ) D

21 Summary We have practised our skills involving Algebraic Fractions We have followed the same rules which we use for numerical fractions We have also learnt how to deal properly with remainders in Algebraic division