Question

How do I use the conjugate zeros theorem?

Answer 1

Hint: In this question, we have been asked how the conjugate zeros theorem is used. First, we will read about what this theorem is about. Then, explain how it can be used to solve questions by giving an example.

Complete step-by-step solution:
In order to be able to use conjugate zeros theorem, we should first know what it is.
What is conjugate zeros theorem?
The conjugate zeros theorem states that if a polynomial, with real coefficients, has one of its zeroes as a complex number$a+bi$ , then the complex conjugate of this number, which is $a-bi$ is also the zero of this polynomial. This is also called Conjugate Pair Theorem.
In other words, we know that - a polynomial of degree ${}^{\prime }{n}^{\prime }$ has ${}^{\prime }{n}^{\prime }$ zeros. If in this polynomial, the coefficients are real, then its imaginary zeros will always exist in pairs, and further, always in conjugate pairs.
For example: Let us take an equation, $f\left(x\right)=x^2-6x+13$ .
Since it is a quadratic equation, we know that there will be two zeroes. Let us find out its zeros using quadratic equations. We know that $a=1,b=-6,c=13$ . Putting these values in the quadratic formula,
$\Rightarrow x={\displaystyle \frac{-\left(-6\right)\pm \sqrt{{\left(-6\right)}^2-4\left(1\right)\left(13\right)}}{2}}$
Simplifying this, we get,
$\Rightarrow x={\displaystyle \frac{6\pm \sqrt{36-52}}{2}}$
$\Rightarrow x={\displaystyle \frac{6\pm \sqrt{-16}}{2}}$
We know that $\sqrt{16}=4$
$\Rightarrow x={\displaystyle \frac{6\pm 4i}{2}}=3\pm 2i$
Hence, we can see that in this question also, we got a pair of conjugate complex zeros.
We can say now that if we are given one complex zero of a polynomial, we can find the other zero, without any calculation.

Note: Sometimes, in an entrance/aptitude exam, a question is asked related to this topic. It goes like–
‘If in a polynomial, with degree 7, one zero is complex, then how many maximum zeros can be real?’
Now, as we read above, complex zeros always exist in pairs. So, if one zero is complex, it is mandatory for another zero to be complex as well. Apart from these two, there can be more complex zeros but it is only a possibility, no concrete answer can be found. So, maximum 5 zeroes can be real.
But, what about minimum?
Minimum, only 1 can be real as this particular polynomial will have 7 zeros (as degree=7) and maximum only 6 zeros can be complex as they exist in pairs.
Hence, we can conclude that, if the degree of a polynomial is odd, then it will have at least one real zero.