Do not worry about factoring anything like this. So, before we get into that we need to get some ideas out of the way regarding zeroes of polynomials that will help us in that process.
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This example leads us to several nice facts about polynomials. Here is the first and probably the most important. There is one more fact that we need to get out of the way. There are only here to make the point that the zero factor property works here as well.
This will be a nice fact in a couple of sections when we go into detail about finding all the zeroes of a polynomial. We can go back to the previous example and verify that this fact is true for the polynomials listed there.
Example 1 Find the zeroes of each of the following polynomials. Due to the nature of the mathematics on this site it is best views in write a polynomial given the zeros mode.
Those require a little more work than this, but they can be done in the same manner. This is a great check of our synthetic division.
To do this we simply solve the following equation. Again, if we go back to the previous example we can see that this is verified with the polynomials listed there.
So, if we could factor higher degree polynomials we could then solve these as well. Also, recall that when we first looked at these we called a root like this a double root. If you think about it, we should already know this to be true.
Note as well that some of the zeroes may be complex. In each case we will simply write down the previously found zeroes and then go back to the factored form of the polynomial, look at the exponent on each term and give the multiplicity.
So, why go on about this? To do this all we need to do is a quick synthetic division as follows. It is completely possible that complex zeroes will show up in the list of zeroes. Another way to say this fact is that the multiplicity of all the zeroes must add to the degree of the polynomial.
The next fact is also very useful at times. Show Solution First, notice that we really can say the other two since we know that this is a third degree polynomial and so by The Fundamental Theorem of Algebra we will have exactly 3 zeroes, with some repeats possible.
In this section we have worked with polynomials that only have real zeroes but do not let that lead you to the idea that this theorem will only apply to real zeroes. When we first looked at the zero factor property we saw that it said that if the product of two terms was zero then one of the terms had to be zero to start off with.
The zero factor property can be extended out to as many terms as we need. Zeroes with a multiplicity of 1 are often called simple zeroes. This fact is easy enough to verify directly.
In the next couple of sections we will need to find all the zeroes for a given polynomial. We solved each of these by first factoring the polynomial and then using the zero factor property on the factored form. The factor theorem leads to the following fact.
Example 2 List the multiplicities of the zeroes of each of the following polynomials. We will also use these in a later example.SOLUTION: Write the equation of the polynomial function with the given zeros: 3,4,-2 Algebra -> Polynomials-and-rational-expressions -> SOLUTION: Write the equation of the polynomial function with the given zeros: 3,4,-2 Log On.
Factors and Zeros Date_____ Period____ Find all zeros. Write a polynomial function of least degree with integral coefficients that has the given zeros. 9) 3, 2, −2 f. Writing polynomial functions with complex zeros How do you write a polynomial function (f) of least degree that has rational coefficients, a leading coefficient of one, and the zeros 5, 5, and 4+i?
Algebra 2 Writing Polynomial Functions Complex Zeros. For a polynomial, if #x=a# is a zero of the function, then #(x-a)# is a factor of the function. We have two unique zeros: #-2# and #4#.
However, #-2# has a multiplicity of #2#, which means that the factor that correlates to a zero of #-2# is. You can put this solution on YOUR website! Write the polynomials having the following zeros: 1) -1, 1, 6 If the polynomial has these zeros, then you can write: x = -1, so x+1 = 0 x = 1, so x-1 = 0.
Video: Using Rational & Complex Zeros to Write Polynomial Equations In this lesson, you will learn how to write a polynomial function from its given zeros. You will learn how to follow a process that converts zeros into factors .Download