Is 0 a constant polynomial?
To answer to questions like this one all you need to do is find a definition of polynomial and check it. Here’s a definition from Wikipedia: Polynomial. That definition is quite standard, so when most people use the term polynomial, this definition describes what objects they are talking about.The zero function satisfies that definition and is, therefore a polynomial. The degree of a polynomial in one variable is the exponent of the largest additive term with non-zero coefficient when the polynomial is in standard form. Interestingly, the zero function is the only polynomial without a degree. Other constants are polynomials of degree zero, but zero has no degree.You might have asked this question because the degree is so closely tied to the notion of polynomials that it seems odd that zero can be a polynomial and not have a degree. If that is the origin of your question, then let me say that this idea is certainly not a bad thought. In fact, if the degree were the centrally important property of polynomials, then a better definition would probably be one that excludes zero.However, as far as I can tell, the most important aspect of polynomials is that they form a Vector space. A vector space MUST have a zero vector, and the zero function is the only useful candidate to be that zero vector. If you are interested in WHY it is so important that polynomials are a vector space, you may well need a couple of semesters of linear algebra and real analysis, and maybe some numerical analysis of dynamical systems as well for the applications.
How to write this is standard form?
The given zeros mean that the factors are (x - 2) , (x + 1) , and (x - 1) . Multiply those out to get one of the possible polynomials that has those zeros. (x - 2) * (x + 1) * (x - 1) = (x² - x - 2) * (x - 1) = x³ - x² - x² + x - 2x + 2 = x³ - 2x² - x + 2 ----------------- Here's a graph of that result showing that it is a valid solution : http://s1164.photobucket.com/albums/q561...
How to form polynomial with Degree 4; zeros 4+4i ; 1 multiplicity 2?
I'll assume your polynomial must have real coefficients. If that's the case, then we know complex roots come in conjugate pairs. Therefore, 4 - 4i is also a root. Next, we also know that if r is a root, then (x - r) is a factor of the polynomial. Thus, we can write the polynomial as: f(x) = k(x - 4 - 4i)(x - 4 + 4i)(x - 1)^2 Where k is any arbitrary constant; we normally just choose k = 1 to make it easy. Finally, multiply out all the factors to get the polynomial in standard form.
Writing a polynomial in standard form?
If you don't care what the polynomial looks like, the simplest one is: x - 5 - √2 But if you want a polynomial with rational coefficients, then 5 - √2 must also be a zero. Use the factor theorem to convert the two zeros into factors: (x - 5 - √2) (x - 5 + √2) Multiply the two factors together: x^2 - 10x + 23
How would you factor a prime polynomial with imaginary and complex numbers?
By “prime polynomial”, I assume you mean a polynomial irreducible over the rationals, that is, one that can’t be factored into smaller polynomials with integer or rational coefficients. By “factor”, I assume you mean find its roots.Algorithms for factoring polynomials are implemented in computer algebra system (CAS) like Mathematica or Maxima.If the polynomial is of degree 4 or less, its roots can be expressed in terms of radicals, giving a big, ugly, and messy formula. (The roots of most 5th degree and higher polynomials cannot be expressed as radicals even in principle.) That is rarely of much use, even using a CAS. Sometimes people think that evaluating the symbolic solution numerically is a good way to get numerical answers. Usually, though, it is actually slower and less accurate than using a good numeric method.If you’re lucky, you may be able to decompose it into a composition of polynomials, which you can then solve separately and re-compose.There are various other special polynomials, like the cyclotomic polynomials, whose roots are easy to find.Beyond that — and frankly often even if one of these techniques works — your best bet is to find numerical approximations to the polynomial roots. There are various efficient root-finding algorithms for both real and complex roots, and can calculate roots to any desired precision. These, too, are implemented in all the CAS’s.
Why does studying polynomials is important?
Polynomials are algebraic expressions that add constants and variables, and are formed using power functions.They are much more easy to handle than other complex functions.Quick Roots, easy integration and differentiation.Why learn them :Polynomials can be used to plot complex curves that decides the path of missile trajectories or a roller coaster or model a complex situation in physics experiment.Polynomial modeling functions can be even be used to solve questions in chemistry and biology.So I think they are pretty important.
Let f be fourth-degree polynomial with the zeros -2,6,2i,-2i?
how many distinct linear factor does f(x) have? how many distinct solutions does f(x)=0 have? how many x-intercepts does the graph have? write a polynomial in factored form! then in standard form plz!! im sorry im in summer school taking a class i havt takin before and the work they gave me has nothin gto do with this!! so plz help thanx =)
Help me write a polynomial function with the given zeros?
Have you heard of root form? I.e. y = a(x - r1)(x - r2)(x - r3)...(x - rn). Same thing here. (x - (-5 - sqrt(7)))(x - (-5 + sqrt(7)))(x - (-7 - 6i))). Now, this wont give a polynomial. This is because we have an imaginary root there, however we know imaginary roots come in conjugate pairs, i.e. if - 7 - 6i is a root, then -7 + 6i is a root (just like for radicals). Thus, we have (x - (-5 - sqrt(7)))(x - (-5 + sqrt(7)))(x - (-7 - 6i)))(x - (-7 + 6i))). As for simplifying this, just do (a - b)(a + b) = a^2 - b^2. I.e., x^2 - x(-5 - 5 + sqrt(7) - sqrt(7)) + (25 - 7) = x^2 + 10x + 18. Likewise, x^2 - x(-7 - 7 + 6i - 6i) + (49 - (-36)) = x^2 + 14x + 85. I.e. (x^2 + 14x + 85)(x^2 + 10x + 18). You can expand this yourself to get the quartic polynomial. Really, the only key thing about this problem is to realize roots come in conjugate pairs.
What is a polynomial hash function for a string?
Anything you store in a computer is a number. Even what appears as a string is stored as a sequence of 1's and 0's and can be interpreted as a number. Polynomial hash functions apply some form of polynomial division on that number (i've forgotten the exact calculations). You can read about CRC in Wikipedia. It is a very common Polynomial Hash Function that is in use.
4,-8, and 2+5i Write a polynomial function of minimum degree with real coefficients whose zeros include those?
(x-4)(x+8)(2-5i)(2+5i)