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Using the Conjugate Zeros Theorem - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

There are different methods for finding the zeros of an expanded polynomial, one of which uses the conjugate zeros theorem. If we are given an imaginary zero, we can sometimes use the conjugate zeros theorem to factor the polynomial and find other zeros. The conjugate zeros theorem says that if a polynomial has one complex zero, then the conjugate of that zero is a zero itself.

I want to talk about the zeros of polynomial function and let's start by looking at 3 examples of quadratic functions which are polynomials and my 3 examples, remember the graph of a quadratic is a parabola and these 3 examples show one of the things about polynomial functions the zeros you can have two real zeros for a quadratic function, one real zero or none. And I want to investigate that a little bit further for example for this function f of x equals x squared minus 6x-7 you can factor this in order to find the zeros right? The x squared means we're going to have an x and an x and the 7 means I'm going to have a 1 and 7. So I just need to figure out whether I need a plus or minus here and here in order to get the minus 6. Well one has to be plus and one has to be minus to get the minus 7. So let's make this minus and this plus and I'll get x-7x is minus 6x so that works, zeros for this guy our negative 1 and 7 the zeros of these two factors.
Now let's look at this guy, the one that sort of touches the x axis and bounces off right? This is a perfect square and factors is x-3 squared. And so it's zeros are just three, well because x-3 is a factor twice, we say that 3 is a zero of multiplicity 2. And finally what are the zeros of this polynomial, and I mean across the x axis but it does have zeros, there are numbers that will make this polynomial zero but they're imaginary.
Let's use the quadratic formula to find them, so we need x=-b, 6 plus or minus the square root of -b squared which is 36 minus 4ac so minus 4 times 13, which is negative 52 all over 2a over 2. So I have 6 plus or minus, this is -16 and the square root of -16 if 4i over 2. This becomes 3 plus or minus 2i, these are the two zeros, zeros of h, 3 plus or minus 2i. So when you look at the results here you see that the function f had 2 real zeros, this one has two real zeros if you count multiplicity and this one has two imaginary zeros.
Let's take a look at a theorem that's going to be true for all polynomials not just quadratics. A degree n polynomial has n zeros counting multiplicity. Always n so like a degree 5 polynomial will have 5 zeros if you count multiplicity. The conjugate zero is another important one. If polynomial function has all real coefficients and most of the ones we study do it's imaginary zeros come in conjugate pairs. Just to show you this was an example of that. This function h of x, this has all real coefficients 1 -6 and 13 are real numbers and these 2 zeros 3 plus and minus 2i those are conjugate pairs so the imaginary zeros came in conjugate pairs. That always happens as long as your polynomial has all real coefficients. So let's take a look at an example, here I have a third degree polynomial right, a degree 3 polynomial by this theorem is going to have 3 zeros and if I know that f of 5+i=0 then I know that 5+i and 5-i are zeros. So that's how I use these 2 theorems to analyze the zeros of a polynomial and I'll do a little bit more with that in a future example.

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