JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Polynomial

Algebraic expression containing many terms of the form \[c{{x}^{n}},\,\,n\] being a non-negative integer is called a polynomial. i.e., \[f(x)={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+{{a}_{3}}{{x}^{3}}+......+{{a}_{n-1}}{{x}^{n-1}}+{{a}_{n}}{{x}^{n}}\], where \[x\] is a variable, \[a>0,\,(<0)\] are constants and \[{{a}_{n}}\ne 0\].

 Example : \[4{{x}^{4}}+3{{x}^{3}}-7{{x}^{2}}+5x+3\], \[3{{x}^{3}}+{{x}^{2}}-3x+5\].

(1) Real polynomial

\[f(x)={{a}_{0}}+{{a}_{1}}x+\]\[{{a}_{2}}{{x}^{2}}+\]\[{{a}_{3}}{{x}^{3}}+\]\[.....+{{a}_{n}}{{x}^{n}}\]

is called real polynomial of real variable x with real coefficients.

Example: \[3{{x}^{3}}-4{{x}^{2}}+5x-4,\,{{x}^{2}}-2x+1\]

etc. are real polynomials.

(2) Complex polynomial

\[f(x)={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+{{a}_{3}}{{x}^{3}}+\]\[......+{{a}_{n}}{{x}^{n}}\]

is called complex polynomial of complex variable \[x\] with complex coefficients.

Example: \[3{{x}^{2}}-(2+4i)x+(5i-4),\,{{x}^{3}}-5i\,{{x}^{2}}+(1+2i)x+4\] etc. are complex polynomials.

(3) Degree of polynomial : Highest power of variable x in a polynomial is called degree of polynomial.

Example: \[f(x)={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+......+{{a}_{n-1}}{{x}^{n-1}}+{{a}_{n}}{{x}^{n}}\] is a \[n\] degree polynomial.

\[f(x)=4{{x}^{3}}+3{{x}^{2}}-7x+5\] is a 3 degree polynomial.

A polynomial of second degree is generally called a quadratic polynomial. Polynomials of degree 3 and 4 are known as cubic and biquadratic polynomials respectively.

(4) Polynomial equation : If \[f(x)\] is a polynomial, real or complex, then \[f(x)=0\] is called a polynomial equation.