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Pc 2.3/2.4 Real and Complex Zeros of Polynomials

To understand zeros of polynomials it is important to understand where they come from.  We can quickly find the zeros of a polynomial when we know the factors.  When a polynomial is in factored form we can use the zero product property to find the zeros.  However it is also important to understand that factors not only help us find the zeros but can also help us find other factors and zeros.  Remember if we have a factor then that means that the factor evenly divides into a polynomial giving us another factor(the quotient).  In these two chapters we will rely heavily on synthetic division but we will also need to know how to divide a factor in a polynomial when the factor in no linear.  Meaning not in the for (x-k) where (x-k) is a factor and x=k is the zero or x-intercept. To use synthetic division you must have a factor in the form (x-k).

Dividing two polynomials using long division Part 1

Dividing two polynomials using long division Part 2

Dividing two polynomials using synthetic division

Now that we know how to divide we will move on to finding all of the zeros of a polynomial when a polynomial is not in factored form, nor can it be factored. To do this our first step will be to identify all of the real rational zeros. To do this we will need to apply the rational zero test.

Apply the Rational Zero Test to determine the number of possible real rational solutions

This helps us in narrowing down the field of what could be our possibly zeros. However some problems will tell you what one of the zeros, factors or roots may be. In that case you can jump right in to synthetic division.

Finding the zeros of a polynomial when given one factor, zero or root

If we are left out to dry and only given an equation we have two options. Graph the polynomial using graphing technology and find the real rational zeros on our own. Remember the zeros are the x-intercepts. We can then use that information to start dividing and finding the zeros. If we do not have access to technology then we must test each rational zero till we find one that is a zero. I will show you examples of both

Using the rational zero test, to test and find all of the zeros of a polynomial by synthetic division

Using the rational zero test and synthetic division to find all of the zeros, rational, irrational and complex of a polynomial

Since we are now talking about complex zeros remember that the Fundamental Theorem of Algebra tells us that the number of zeros, real rational, real irrational, and complex is equal to the degree of the polynomial. Therefore we can have complex zeros but they also come in conjugate pairs like irrational zeros because we find them by taking the square root of a negative number where irrational come from taking the square root of non square positive numbers. When solving by taking the square root you have to include plus and minus. If we have complex zeros that means we can also write the equation of a polynomial given irrational and complex zeros.

How to write the equation of a polynomial given complex zeros

Lastly to bring all of this together we will discuss Descartes Rule of signs. Allowing my videos to explain the details, Descartes Rule of signs is a great way for us to check our answer when finding the zeros of a polynomial by giving us the number of real positive, real negative and complex zeros of a polynomial

Apply Descartes Rule of Signs to determine the number of real positive, real negative and complex zeros of a polynomial


 

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