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.
Write each function in factored form, if possible, using integer coefficients.?
a.y(x) = x^2 + 17x + 60 b.k(p) = p2 + 5p + 11 Input positive integers as your answers, if possible. Arrange the numbers increasingly, when possible. Type zeros in blanks if the function can not be factored using integer coefficients. a.y(x) = (x + __)(x +__ ) b.k(p) = (p + __)(p - __) I know letter be would be (0,0) but not a. Help :D
How do I find all the roots of the quartic polynomial [math]x^4+x^3+14x^2+16x-32[/math]?
It depends on your personal context. Sometimes we encounter problems like this in school, sometimes in recreational mathematics. We rarely see them in the working world, at least as I have experienced it.If you saw this in school then you should look carefully through your class notes and textbooks for advice about what techniques you are expected to be able to use. I recommend this approach particularly in the case of quartic polynomials because solving them in the general case without using computers is a gruesome process that is involves numerous steps that are open to error. However, the polynomial you have presented in the question yields easily to some attacks that you have probably been taught.See How to Solve a Quartic | The Classroom | Synonym for the steps involved in solving quartics in general. (Think how miserable life would be if we had to do that on examinations.) Others here have suggested other approaches.As a programmer I would be inclined to solve the equation in this way:The advantage of using computer code is that I could use the roots in other parts of the same code.
How does the zero of the polynomials help to find any equation?
The idea is that by bringing all terms to one side of the equation, you get an expression that equals 0. First, you need to factor the polynomial you obtain into a product of terms. Since the terms are real or complex numbers, a product of terms equal to 0 must contain one term that by itself equals 0. That term could be any of the factors or group of factors. So, the roots are all of the possible values the unknown (say x) is permitted to have. These are all the possible solutions, the values of x consistent with the equation that generated the polynomial.
How do I factor this equation using decomposition? 6x^2+11x+3
Thank you for the A2A!So, decomposition is a way of factoring that allows us to break apart complex trinomials and make them easier to factor.The first step in decomposition would be, in the form ax^2 +bx +c, find which two numbers add to b, and multiply to a*c.In the case of 6x^2 +11x + 3, we need two numbers that add to 11 and multiply to 18.Quickly cycling through the factors of 18 leaves us with 9 and 2 (multiply to 18, add to 11).From there, we break the middle term into these two parts:6x^2 +11x + 3 becomes:6x^2 + 9x +2x + 3Then, we can factor each half by getting the gcf of the first two terms, and the gcf of the second two terms. 6x^2 + 9x +2x + 3becomes:3x (2x+3) + 1 (2x+3)Which then can be factored again by taking out the 2x+3, giving us a final answer of (3x+1)(2x+3)I really hope this helps! Please let me know if you have any questions!Conner D
What is the correct way to factor on a graphing calculator?
To factor a polynomial, assuming you’re on a TI calculator, you want to use the poly function for the 85/86 line, which is on the main screen. You drop in your coefficients and solve.For the Ti-83 line (83/84/etc)For the 89/92/NSPIRE, you can Texas Instruments