C3 - Algebraic Fractions

Question 1

(1/2x + 1) /

1/3x +2/3

Answer 1

Times the numerator and the denominator by 6
Factorise

= 3/2

Question 2

(x - 1/x) /

x + 1

Answer 2

Multiply the num and den by x to remove the fraction
Factorise to get a common factor
Cancel down

= x-1 /
x

Question 3

(1/2x^2 + x - 4) /

1/4x^2 + 3/2x + 2

Answer 3

Times by lowest common multiple = 4
Factorise into double brackets
Cancel down

= 2x - 4 /
x + 2

Simplify again

= 2(x -2) /
x + 2 make sure you can’t put it into brackets anymore

Question 4

(x^2 - 5x - 6) /

(1/3x - 2)

Answer 4

Multiply by lcm 3 but with the numerator, don’t times out leave as:

3(x^2 - 5x - 6) /
x - 6

Factorise

3(x-6)(x+1) /
(x-6)

= 3(x+1)

Question 5

3/5 x 5/9

Answer 5

Cancel down the 3 and 9 to 1 and 3
Cancel down the 5 and 5 to 1 and 1

1/1 x 1/3 = 1/3

Question 6

(x + 1)             3
    2      x   (x^2 - 1)

Answer 6

Factorise (x^2 - 1) into (x - 1)(x + 1)
Cancel down the (x + 1) into 1

1/2 x 3/(x-1)

= 3
2(x-1)

Question 7

(x + 2) (3x + 6)

x + 4) / (x^2 - 16

Answer 7

Flip the second fraction
Factorise
Cancel down the x + 4 into 1

= x + 2 x - 4
1 x 3x + 6

Multiply into one fraction keeping the brackets

(x + 2)(x + 4)
3(x + 2)

= (x - 4)
3

Question 8

x^2 + 2xy + y^2 / 2 x 4/ (x - y)^2

Answer 8

Factorise first expression into (x + y)^2

Cancel down the 2 to 1 and the 4 to 2

Times out to give:

2(x + y)^2 /
(x - y)^2

Question 9

What’s the rule for adding and subtracting numerical and algebraic fractions?

Answer 9

Find LCM
Divide the LCM by the denominator of each fraction
Times the whole fraction by that value

Eg:

1/3 + 3/4

LCM: 12

12/3 = 4      4(1/3) = 4/12
12/4 = 3      3(3/4) = 9/12    

Now add = 13/12

Question 10

3/(x + 1) - 4x/(x^2 - 1)

Answer 10

Expand (x^2 - 1) to give (x + 1)(x - 1)

LCM: (x + 1)(x - 1)

Divide the LCM by both fraction denominators

1st gives (x - 1) 2nd gives 1

Times respective fractions to give:

3x - 3/ - 4x/
(x - 1)(x + 1) (x - 1)(x + 1)

= - x -3/
(x + 1)(x - 1)

Question 11

1/3(x + 2) - 1/2(x + 3)

Answer 11

Times out
Put each fraction over 6 (group together the algebraic and numerical fractions usually)
Simplify and put into ONE fraction

= - x -5 / 6

Question 12

(x + 2) / (x^2 - x - 12) - (x + 1) / (x^2 + 5x + 6)

Answer 12

Factorise the denominators
 LCM: (x + 3)(x + 2)(x - 4)
Times the numerators by the missing denominator factor
Expand 
Simplify

= 7x + 8/ (x + 3)(x + 2)(x - 4)

(if you didn’t get this, remember that you need to use the - sign separately to the second expression)

Question 13

Why does the remainder go over the divisor for the final remainder total

Answer 13

Because you are dividing by x -3 so the remainder goes over it

Question 14

What are the main things to remember when cancelling down fractions

Answer 14

Multiply out fractions by the LCM
Factorise, factorise, factorise
Cancel out numbers that go into each other

Question 15

What are the main things to remember when multiplying and dividing fractions

Answer 15

When multiply do it across the lines, cancel down numbers first if you can
With dividing, flip the second fraction and continue as a multiplication

Question 16

What are the main things to remember when adding and subtracting fractions

Answer 16

Make the denominators the same
KEEP AS ONE FRACTION
Factorise where possible and cancel down
Find the LCM of the denominators and times the numerators by what’s missing

Question 17

What are the main things to remember when using long division or the remainder theorem

Answer 17

It produces a mixed number fraction

The remainder goes over the divisor next to the quotient to form the mixed number