Complex number technique: Multiply by conjugate

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.

complex number nov 95

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

Complex number technique: ‘Taking out half power’

‘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 w5=1

Hence show that the roots of the equation Tex2Img_1405327571 are Tex2Img_1405327808 where Tex2Img_1405327849

Solution

Tex2Img_1405327984

Therefore

Tex2Img_1405328069

2w-wz=2+z

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

Tex2Img_1405328177

Tex2Img_1405328778

Tex2Img_1405329158 (Taking out half power)

Tex2Img_1405329321

=Tex2Img_1405327808 where Tex2Img_1405327849 (proved)

Example 2

complex 2015

Example 3

Capture

‘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

Using GC in Complex Numbers

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  Tex2Img_1404187817 has a complex root 1-i.

Solution

Since 1-i is a root,

Tex2Img_1404188152

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

2+k = 0

k=-2

Example 2

Find the 2 roots of Tex2Img_1404188506

Solution

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

Tex2Img_1404188717

Using the GC to evaulate the above,

z = 1+2i or -7i

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