1. Manchester University Revision Day
Algebra for Edexcel C2
Manchester University
Revision Day

2. Algebra What you need to know Algebraic division
Factor theorem and remainder theorem
The exponential graphs
Logs, indices and their laws
Solving equations and inequalities with an unknown power

3. Formulae you need to learn
Remainder when a polynomial f(x) is divided by (x-a) is f(a)
Laws of logs

4. Algebraic division
Write the x +3 on the outside and the first term on the inside x x3 3
Fill in the first column

5. How many more x2 do we need?3
3x2

6. Finish off the table x2 x x3
4x2 3
3x2

7. x2 4x
-12 x x3
4x2
-12x 3
3x2
12x
-36
Now factorise the quadratic factor

8. x2 4x
-12 x x3
4x2
-12x 3
3x2
12x
-36

9. Working with cubics
Write the x -2 on the outside and the first term on the inside x
2x3 -2
Fill in the first column

10. Working with cubics
How many more x2 do we need?
2x2 x
2x3 -2
-4x2

11. Working with cubics
Finish off the table writing an extra number in a box to the right for the remainder
2x2 x
2x3
7x2 -2
-4x2

12. Working with cubics 2x2 7x +7 x 2x3 7x2 -2 -4x2 -14x -14 18
Quotient is 2x2 +7x +7 and remainder 18
WARNING: DON’T use this method if the question says “use the remainder theorem” to find the remainder. You’ll get NO MARKS!

13. Practise this for yourself
– all of these cubics can be written as the product of linear factors
(x - 5 ) is a factor of x3 - x2 - 17x - 15
(x + 3 ) is a factor of 2x3 + 4x2 - 18x - 36
(3x + 1) is a factor of 3x3 + 10x2 + 9x + 2
Some calculators have the facility to solve quadratic and cubic equations. That can really help here

14. Answers (x - 5 ) is a factor of x3 - x2 - 17x – 15

15. Factor and remainder theorem
The remainder when a polynomial f(x) is divided by (x-a) is f(a).
In particular, if (x-a) is a factor, f(a) = 0
The remainder when a polynomial f(x) is divided by (ax+b) is

16. Example 3
Show that (x+2) is a factor of and solve the equation f(x)=0.
So (x+2) is a factor

17. Example 4
If (x+4) is a factor of find k

18. Example 5 Find the remainder when is divided by
Choose the value of x which makes the bracket zero
Remainder = (You can check this by division)

19. Example 6 The remainder when is divided by (x+1) is 11. Find a.
I have seen students do this by working forwards and backwards through the grid method.
This method works best if there are two unknowns and two statements.

20. Exponentials and logs

21. Exponentials and logs

22. Powers of numbers less than 1

23. Powers and logs
The logarithm of a number is the power of the base you need to make the number.
If no base is given, it means base 10

24. Examples
Fill in the blanks in each statement and rewrite using logs.

25. Using the laws of logs

26. Example 7
Write as a single log

27. Example 8
Write as a single log

28. Example 9
Write
in the form

29. Solving equations and inequalities with an unknown power
Example 10 Solve the equation 2x = 2√2
Or you could use logs

30. Check log5 is positive before dividing.
Example 11
Solve the inequality 5x < 120
OR solve the equation…
… and use common sense to decide < or > in your answer
Check log5 is positive before dividing.
If the power is attached to a number less than one, the log has a negative value and the inequality reverses when you divide..

31. Check your answer by substitution
Example 12
Solve the equation
Check your answer by substitution

32. Example 13 Solve the equation Notice that is the same as
The equation is quadratic in

33. Summary What you need to know Algebraic division
Factor theorem and remainder theorem
The exponential graphs
Logs, indices and their laws
Solving equations and inequalities with an unknown power