Algebraic fractions

1. Basic simplifying 2

In an algebraic fraction, you can only cancel things which are connected by MULTIPLICATION signs For example, in there are 4 terms – the 7b , the top a, the 6 and the bottom a. The ONLY signs between any of these anywhere are the × signs. So, if there are two things the same in top and bottom, we can cancel them. So cancel the a’s. However, in the presence of any + or – signs prevents the possibility of cancelling and so these three expressions would have to remain unsimplified.

Impossible due to + & – connecting terms in top & bottom Now put yourself to the test. Which of these can be simplified? You may need to insert × signs No + or – anywhere No + or – anywhere Impossible due to + & – connecting terms in top & bottom Impossible due to + connecting terms in top No + or – anywhere Note the + does not connect the main terms. It is part of the term (a + 5) and so can be ignored! Can’t cancel the 5s because of the + which now connects terms!

2. Multiplying & Dividing Multiplying Algebraic Fractions.......... Factorise numerator & denominator first Cancelling allowed (but one term being cancelled has to be in the top and the other in the bottom). Dividing Algebraic Fractions.......... Invert second fraction before you do anything! Then proceed as you would for a multiplication!

Inverting the 2nd fraction Factorising Cancelling

On first glance it looks like this can’t be done because of the minus On first glance it looks like this can’t be done because of the minus. However if we factorise the numerator This now has a multiplication connecting the terms in the numerator and so an a can be cancelled As you get more confident with these, you can go straight from the first expression to the answer by cancelling an a from all 3 terms, avoiding the need to factorise

3. Adding & Subtracting Cannot do this unless the denominators are the same Once the denominators are the same, then add only the numerators.

Top & bottom of first fraction are multiplied by 4 Bright idea! At the start, always bracket any numerator that has 2 or more terms! As this sum is a + , we can’t do it until the denominators are the same. Choose 20 Top & bottom of first fraction are multiplied by 4 Top & bottom of second fraction are multiplied by 5

As this sum is a “ – “ , we can’t do it until the denominators are the same. Choose 6 Note the sign change!

Common denominator is xy. Multiply top and bottom of 1st fraction by y Multiply top and bottom of 2nd fraction by x

Extension work

Mult top & bottom of 1st fraction by (x – 3) Common denom (x + 1)(x – 3) Mult top & bottom of 1st fraction by (x – 3) Mult top & bottom of 2nd fraction by (x + 1) Now both denoms are the same so we can do the subtraction Note the change of sign on the 5 in the top

These denominators look different but in fact they are negatives of each other e.g. 3 – 7 is the negative of 7 – 3 The strategy here is to reverse the denominator and change the sign in front. Both denoms are now the same

Since x divides into x2, choose x2 as the common denominator. Mult. Top & bottom of first fraction by x. Leave 2nd fraction as is. Now both denoms are the same so we can do the subtraction

Since (x + 3) divides into (x + 3)2 choose (x + 3)2 as the common denominator. Mult. Top & bottom of 2nd fraction by (x + 3) Leave 1st fraction as is. Now both denoms are the same so we can do the addition

Changing signs within fractions..... changing the sign in front of all terms = rewriting the bottom putting the positive (w) in front of the – t = You’re allowed to take the single minus from the top and put it out the front of the whole fraction =

Rewrite these in an alternative form.... Both equal 2 Both equal – 2

Realising (x – 3) and (3 – x) are negatives of each other, reverse the 2nd denom and change the sign To bring 2nd denom into line with 1st we mult top & bottom by (x – 3)

Quite complex!! Factorise denoms to find common factors LCD will be (x – 3)(x + 3)(x + 1) Expand & collect terms in the top

Problem Solver Questions + Worked Solutions

Toughies…Solve these Solve…. a. a. b. b. c. c.

Back to questions Need to ADD powers on LHS Rewrite all numbers as powers of 2 Need to ADD powers on LHS And as both sides are of the form 2POWER we can equate the powers…. Multiply all four terms by LCD x(x – 1) Expand & clean up (x2 terms will cancel!) Back to questions

Ans x = 5 and y = – 2 ½ x – 2y = 10……(1) 3x + 4y = 5…….(2) 5x = 25 Equate powers x – 2y = 10……(1) 3x + 4y = 5…….(2) NOTE Powers of 2 in first eqn and powers of 3 in the second. Rewrite. 2  (1) + (2) to eliminate y 5x = 25 So x = 5 Expand brackets From (1) 5 – 2y = 10 y = – 2 ½ Add/subtract powers Ans x = 5 and y = – 2 ½ Back to questions

x = 4.383 or – 1.217 Back to questions