Polynomials in One Variable
The general form of a polynomial in one variable is represented as p(x) = a_{n} x^{n} + a_{n - 1} x^{n - 1 } + .... a_{1} x^{1} + a_{0}
Where:
* a_{n} , a_{n - 1} , .... , a_{1} , a_{0} are real numbers,
* The exponents of each terms are whole numbers and
* The highest exponent, n, is the degree of the polynomial, if an a_{n} ≠ 0.
If p(x) is a polynomial and k is real number and if p(k) = 0, then k is called the zero of the polynomial.
Two methods to determine the zero of polynomial are:
* Trial and error
* Equating polynomial to zero
Remainder Theorem
The remainder theorem states that if p(x) is a polynomial in x and p(x) is divided by (x – a), then the remainder is p(a).
If a polynomial p(x) is :
* Divided by (x + a), then the remainder is p(-a).
* Divided by (ax – b), then the remainder is p( \frac{b}{a} ).
* Divided by (ax + b), then the remainder is p( \frac{- b}{a} ).
Factor Theorem
* Factor theorem states that “p(x) is a polynomial and a is real number, if p(a) = 0,
Then x – a is factor of p(x).”
* If p(x) is a polynomial and x – a is factor of p(x), then p(a) = 0.
Factorization of Polynomials Using Algebraic Identities
* (a + b)³ = a³ +b³ + 3ab (a + b)
* (a – b)³ = a³ – b³ -3ab (a – b)
* a³ + b³ = (a + b) (a² – ab + b²)
* a³ – b³ = (a – b) (a²+ ab +b²)
* a³+ b³+ c³ – 3abc = (a + b + c) (a²+ b²+ c² – ab – bc – ca)