MandysNotes

Polynomials

08 March 2014 By In Polynomials

Definition:


A  polynomial function of a real variable \(x\), is a function defined by the sum:
\[ P(x) = \sum_{k=0}^{n}c_k x^{k} = c_0 + c_1 x + c_2 x^{2}... + c_n x^{n}. \]

08 March 2014 By In Polynomials

 A linear function, \(f(x)\), of a real number, \(x\), is defined by two properties:

  • for any real number, \(a\), \(f(ax) = a(f(x));\)
  • for any real number, \(h\), \(f(x+h) = f(x) + f(h).\)
09 March 2014 By In Polynomials

The product of two linear factors

\[ (x - \alpha)(x - \beta)\]

is a polynomial of degree two:

\[(x - \alpha)(x - \beta) = x^{2} - (\alpha + \beta)x + \alpha\beta.\]

Definition:


Polynomials of degree two are also called quadratic polynomials.

 

09 March 2014 By In Polynomials

Definition:

The roots of a polynomial \(P\) are the solutions of the equation:

\[ P(x) = 0. \]

05 July 2014 By In Polynomials

How do we divide polynomials?

Suppose, for example that you are given the problems:


Express \(\frac{x^{2} - 1 } {x - 1 } \) in terms of \(x\).


Find \(\frac{x^{3} + 2x^{2} + x}{x + 1}\) in terms of \(x.\)

What do we do?

NOTRad