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Q: Polynomial whose greatest monomial factor is 1?

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A polynomial.

A quadratic polynomial must have zeros, though they may be complex numbers.A quadratic polynomial with no real zeros is one whose discriminant b2-4ac is negative. Such a polynomial has no special name.

The result is a polynomial q(x) whose order is one fewer than the order of p(x) and a remainder term of the form b/(x + a).

If a polynomial p(x), has zeros at z1, z2, z3, ... then p(x) is a multiple of (x - z1)*(x - z2)*(x - z3)... To get the exact form of p(x) you also need to know the order of each root. If zk has order n then the relevant factor in p(x) is (x - zk)n

A polynomial can be factored if it has a rational root. If f(x) is a polynomial function of x and if there is a rational number p such that f(p) = 0 then f(x) = (x-p)*g(x) where g(x) is a polynomial whose order is one less than the order of f(x). If p = q/r where q and r are integers, then (x - p) = (x - q/r) = (rx - q)/r which is a rational binomial factor. This does not work if p is irrational which is why p must be rational.

Related questions

prime

A polynomial.

If you know one linear factor, then divide the polynomial by that factor. The quotient will then be a polynomial whose order (or degree) is one fewer than that of the one that you stared with. The smaller order may make it easier to factorise.

479 and 958 are two numbers whose greatest common factor is 479, among others.

12 and 24.

4 and 9

8M and 12M2

8 and 16.

90 and 180

10 and 15

16 and 8

24 & 36

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