Algebraic Fractions

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

Teachings for Exercise 1A

Algebraic Fractions 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. In the second example, you cannot just ‘cancel the x’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 Example Questions Divide by the common Factor (4) Divide by the common Factor (4) Factorise the Denominator Factorise the Denominator Divide by the common Factor (x + 3) Divide by the common Factor (x + 3) 1A

Algebraic Fractions 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) Example Questions Multiply the Numerator and Denominator by 6 Multiply the Numerator and Denominator by 6 Factorise Factorise Divide by (x + 2) Divide by (x + 2) 1A

Algebraic Fractions 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. Example Questions Factorise the Numerator AND Denominator Factorise the Numerator AND Denominator Divide by (x + 1) Divide by (x + 1) 1A

Algebraic Fractions 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 Example of a Fraction within a Fraction… You will usually be told what ‘form’ to leave your answer in… Example Questions Multiply the Numerator and Denominator by x Multiply the Numerator and Denominator by x Factorise Factorise Divide by (x + 1) Divide by (x + 1) Split the Fraction up 1A

Teachings for Exercise 1B

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 Example Questions a) b) c) 1B

Algebraic Fractions 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 Example Questions 1 d) 1 e) Factorise 1 1 Multiply Numerator and Denominator 1B

Algebraic Fractions 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. Example Questions a) 1B

Algebraic Fractions 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. Example Questions b) 1 1 1B

Algebraic Fractions 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. Example Questions c) Leave, Change and Flip Factorise 1 1 1 1 Multiply the Numerators and Denominators 1B

Teachings for Exercise 1C

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. Example Questions a) Multiply all by 4 Multiply all by 3 Add the Numerators Add the Numerators 1C

Algebraic Fractions 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. Example Questions Example Questions b) Imagine ‘b’ as a Fraction Multiply all by x Combine as a single Fraction Combine as a single Fraction 1C

Algebraic Fractions c) 1C 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. Example Questions c) Factorise so you can compare Denominators Factorise so you can compare Denominators Multiply by (x - 1) Expand the bracket, and write as a single Fraction Expand the bracket, and write as a single Fraction Simplify the Numerator Simplify the Numerator 1C

Teachings for Exercise 1D

You need to remember how to divide using Algebraic long division Algebraic Fractions x2 + 5x - 2 You need to remember how to divide using Algebraic long division We are now going to look at some algebraic examples.. 1) Divide x3 + 2x2 – 17x + 6 by (x – 3) So the answer is x2 + 5x – 2, and there is no remainder This means that (x – 3) is a factor of the original equation x - 3 x3 + 2x2 – 17x + 6 x3 – 3x2 5x2 - 17x + 6 5x2 - 15x Second, Divide 5x2 by x 5x We then subtract ‘5x(x – 3) from what we have left Third, Divide -2x by x -2 We then subtract ‘-2(x – 3) from what we have left First, Divide x3 by x x2 We then subtract ‘x2(x – 3) from what we started with - 2x + 6 - 2x + 6 1D

You need to remember how to divide using Algebraic long division Algebraic Fractions You need to remember how to divide using Algebraic long division Always include all different powers of x, up to the highest that you have… Divide x3 – 3x – 2 by (x – 2)  You must include ‘0x2’ in the division…  So our answer is ‘x2 + 2x + 1. This is commonly known as the quotient x2 + 2x + 1 x - 2 x3 + 0x2 – 3x - 2 x3 – 2x2 2x2 – 3x - 2 Second, divide 2x2 by x = 2x Then, work out 2x(x – 2) and subtract from what you have left Third, divide x by x = 1 Then, work out 1(x – 2) and subtract from what you have left First, divide x3 by x = x2 Then, work out x2(x – 2) and subtract from what you started with 2x2 – 4x x – 2 x – 2 1D

You need to remember how to divide using Algebraic long division Algebraic Fractions You need to remember how to divide using Algebraic long division Sometimes you will have a remainder, in which case the expression you divided by is not a factor of the original equation… Find the remainder when; 2x3 – 5x2 – 16x + 10 is divided by (x – 4)  So the remainder is -6. 2x2 + 3x - 4 x - 4 2x3 - 5x2 – 16x + 10 2x3 – 8x2 3x2 – 16x + 10 Second, divide 3x2 by x = 3x Then, work out 3x(x – 4) and subtract from what you have left Third, divide -4x by x = -4 Then, work out -4(x – 4) and subtract from what you have left First, divide 2x3 by x = 2x2 Then, work out 2x2(x – 4) and subtract from what you started with 3x2 – 12x -4x + 10 -4x + 16 -6 1D

Algebraic Fractions 19 ÷ 5 = 3 26 ÷ 3 = 8 1D You need to remember how to divide using Algebraic Long Division But, how do we deal with the remainder? 4 The ‘remainder’ is the numerator 19 ÷ 5 = 3 5 5 divides into 19 3 whole times… The ‘divisor’ is the denominator Another way to think of this sum is  19 = (3 x 5) + 4 2 The ‘remainder’ is the numerator 26 ÷ 3 = 8 3 3 divides into 26 8 whole times… The ‘divisor’ is the denominator Another way to think of this sum is  26 = (8 x 3) + 2 1D

Algebraic Fractions 2x2 + 3x - 4 x - 4 2x3 - 5x2 – 16x + 10 2x3 – 8x2 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: 2x2 + 3x - 4 x - 4 2x3 - 5x2 – 16x + 10 2x3 – 8x2 3x2 – 16x + 10 3x2 – 12x Remainder Divisor -4x + 10 -4x + 16 - 6 2x3 - 5x2 – 16x + 10 ÷ (x – 4) = 2x2 + 3x - 4 + x - 4 -6 6 = 2x2 + 3x - 4 - x - 4 1D

Algebraic Fractions x2 + 3x - 3 x - 1 x3 + 2x2 – 6x + 1 x3 – x2 You need to remember how to divide using Algebraic Long Division Write (x3 + 2x2 – 6x + 1) ÷ (x – 1) in the form:  Just do the division as normal… x2 + 3x - 3 x - 1 x3 + 2x2 – 6x + 1 x3 – x2 3x2 – 6x + 1 3x2 – 3x -3x + 1 -3x + 3 -2 Multiply both sides by (x – 1) 1D

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