COMPLEX

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COMPLEX
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Complex zeros or roots of a polynomial could
result from one of two types of factors
Type 1
Type 2
Notice that with either type, the complex zeros
come in conjugate pairs. There are 2 complex
solutions that have the same real term and
opposite signs on imaginary term. This will
always be the case. Complex zeros come in
conjugate pairs.
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So if asked to find a polynomial that has zeros,
2 and 1 3i, you would know another root would
be 1 3i. Lets find such a polynomial by
putting the roots in factor form and multiplying
them together.
If x the root then x - the root is the factor
form.
Multiply the last two factors together. All i
terms should disappear when simplified.
-1
Now multiply the x 2 through
Here is a 3rd degree polynomial with roots 2, 1 -
3i and 1 3i
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Lets take this polynomial and pretend we didnt
know the roots and work the other direction so we
can see the relationship to everything weve
learned here.
?
1, 2, 4, 5, 10, 20
Possible rational roots
1
By Descartes Rule there are 3 or 1 positive real
zeros (3 sign changes in f(x)) and no negative
real roots (no sign changes in f(-x)).
Lets try 2(synthetic division)
Now put variables back in and factor or use
quadratic formula
So there was one positive and two imaginary roots
By quad formula
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Use the given root to find the remaining roots of
the function
Since 3i is a root we also know that its
conjugate -3i is also a root. Lets use
synthetic division and reduce our polynomial by
these roots then.
So the roots of this function are 3i, -3i, 1/3,
and -2
Put variables in here set to 0 and factor or
formula this to get remaining roots.
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Acknowledgement I wish to thank Shawna Haider
from Salt Lake Community College, Utah USA for
her hard work in creating this PowerPoint. www.sl
cc.edu Shawna has kindly given permission for
this resource to be downloaded from
www.mathxtc.com and for it to be modified to suit
the Western Australian Mathematics Curriculum.
Stephen Corcoran Head of Mathematics St
Stephens School Carramar www.ststephens.wa.edu.
au