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

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

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

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

How many more x2 do we need? 3 3x2

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

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

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

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

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

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

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!

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

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

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

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

Example 5 Find the remainder when is divided by Choose the value of x which makes the bracket zero Remainder = 4 (You can check this by division)

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.

Exponentials and logs

Exponentials and logs

Powers of numbers less than 1

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

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

Using the laws of logs

Example 7 Write as a single log

Example 8 Write as a single log

Example 9 Write in the form

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

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..

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

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

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