Write a polynomial equation of least degree with roots

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Write a polynomial equation of least degree with roots

write a polynomial equation of least degree with roots

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In the next couple of sections we will need to find all the zeroes for a given polynomial. 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. To do this we simply solve the following equation.

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Also, recall that when we first looked at these we called a root like this a double root. We solved each of these by first factoring the polynomial and then using the zero factor property on the factored form. 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.

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So, if we could factor higher degree polynomials we could then solve these as well. Example 1 Find the zeroes of each of the following polynomials. Do not worry about factoring anything like this. There are only here to make the point that the zero factor property works here as well.

We will also use these in a later example. Zeroes with a multiplicity of 1 are often called simple zeroes. Example 2 List the multiplicities of the zeroes of each of the following polynomials.

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.

This example leads us to several nice facts about polynomials. Here is the first and probably the most important. Another way to say this fact is that the multiplicity of all the zeroes must add to the degree of the polynomial. We can go back to the previous example and verify that this fact is true for the polynomials listed there.

This will be a nice fact in a couple of sections when we go into detail about finding all the zeroes of a polynomial. Note as well that some of the zeroes may be complex. 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.

It is completely possible that complex zeroes will show up in the list of zeroes. The next fact is also very useful at times. Again, if we go back to the previous example we can see that this is verified with the polynomials listed there.

The factor theorem leads to the following fact. There is one more fact that we need to get out of the way. This fact is easy enough to verify directly. 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.


To do this all we need to do is a quick synthetic division as follows. If you think about it, we should already know this to be true.


So, why go on about this? This is a great check of our synthetic division.Generate polynomial from roots; Generate polynomial from roots. Solving Equations. Quadratic Equations. Polynomial Equations; Rational Equations; Quadratic Equation.

probably have some question write me using the contact form or email me on Send Me . Therefore, a polynomial of even degree admits an even number of real roots, and a polynomial of odd degree admits an odd number of real roots (counted with multiplicity). In particular, every polynomial of odd degree with real coefficients admits at least one real root.

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A root of a polynomial is a number such that. The fundamental theorem of algebra states that a polynomial of degree has roots, some of which may be degenerate. For example, the roots of the polynomial are, 1, and 2.

Finding roots of a polynomial is therefore equivalent to polynomial . How do i write a polynomial function of least degree with intergral coefficients that has the given zeros. the zeros are 3i and 2-i My son is in an algebra 2 class in public HS in CA/5. Polynomial Equation Solver; Polynomial equation solver.

This calculator solves polynomial equations in Polynomial Roots Calculator Was this calculator helpful? formulas and calculators.

If you want to contact me, probably have some question write me using the contact form or email me on Send Me A Comment. Comment: Email (optional). A polynomial function has real coefficients, a leading coefficient of 1, and the zeros 3 and (2 - i).

Write a polynomial function of least degree in standard form. Remember that complex zeros occur in conjugate pairs; therefore, (2 + i) is also a zero. First, let's change the zeros to factors.

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