Polynomials

An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial.
(i) is a polynomial in variable x.
(ii)  is an expression but not a polynomial.
Polynomials are denoted by
Constant polynomial: A polynomial containing one term only, consisting a constant term is called a constant polynomial.  The degree of non-zero constant polynomial is zero.
Zero polynomial: A polynomial consisting of one term, namely zero only is called a zero polynomial.
The degree of zero polynomial is not defined.
Zeroes of a polynomial: Let  be a polynomial. If  then we say that  is a zero of the polynomial  p(x).
Remark: Finding the zeroes of polynomial p(x) means solving the equation p(x)=0.
Remainder Theorem: Let  be a polynomial of degree  and let a be any real number. When  is divided by  then the remainder is
Factor Theorem: Let  be a polynomial of degree  and let a be any real number.
(i) If
(ii)
Factor: A polynomial  is called factor of  divides  exactly.
Factorization: To express a given polynomial as the product of polynomials each of degree less than that of the given polynomial such that no such a factor has a factor of lower degree, is called factorization.

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