Useful in questions like this

Alevel 2014 P2

Solution

SRJC 2018 J2 MYE

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# Category: Complex numbers

## Argument of Real and Purely Imaginary Numbers

## Synthetic division in complex numbers

## Shortcut method for solving 2016 challenging complex number

## Finding the other root in complex numbers

## Challenging Complex number problem 4

## Challenging Complex number problem 3

## Complex number technique: Multiply by conjugate

## Complex number technique: ‘Taking out half power’

## Useful Complex Number property

## Using GC in Complex Numbers

Useful in questions like this

Alevel 2014 P2

Solution

SRJC 2018 J2 MYE

**2016 P1 Q7**

**Solution**

a) Hence part

Common methods to find the other root are long division or compare coef. A faster method is to use the sum of roots taught in additional maths.

b)

Comments from examiner’s report

N2016 P1, 7a

Common methods to find the other root are long division or compare coef. A faster method is to use the sum of roots taught in additional maths.

To simplify the division of complex numbers, two commonly used techniques are “Taking out half power” and “multiply by the conjugate of the denominator”.

This example illustrates the technique of multiply by the conjugate of the denominator.

Taking out half power technique does not work in this case. So we need to know both methods.

‘Taking out half power’ is a useful technique to simplify the division of complex numbers.

**Example 1**

Write down the five roots of the equation w^{5}=1

Hence show that the roots of the equation are where

**Solution**

Therefore

2w-wz=2+z

2(w-1)=z(w+1)

**Example 2**

**Example 3**

‘Taking out half power” only works when the coefficient of both terms are the same in absolute terms. If it isn’t, then have to use another technique: Multiply by conjugate

GC can be very helpful in solving certain problems in Complex Numbers.

**Example 1**

Use the GC to find the real value of k such that has a complex root 1-i.

**Solution**

Since 1-i is a root,

Using TI 84 Plus (OS 2.55) to evaluate the complex numbers, we have

2+k = 0

k=-2

**Example 2**

**Solution**

Since the equation is quadratic, we can use the quadratic formula.

Using the GC to evaulate the above,

z = 1+2i or -7i

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