Polynomials

An algebraic expression of the form, a_n xⁿ +a_n-1 x^n-1 + a_n-2 x^n-2+ .............+ a₁ x + a₀ , where a₀ , a₁ , .............a_n-1 , a_n are real numbers (the set of all rational and irrational numbers forms the set of real numbers), is called a polynomial in x. Only capital letters are used to name polynomials and variables are denoted by small letters. The above polynomial can be denoted by P(x) or Q(x) or R(x) etc. The numbers a₀, a₁, ..........,a_n-1, a_n are called the coefficients in the polynomial. For example, a2 is the coefficient of x² and a_n-1 is the coefficient of x^n-1 and so on.a₀ is called the constant term of the polynomial. Each term of a polynomial has a coefficient and the constant term is the coefficient of x⁰. -2,7,3.4 etc are called constant polynomials. The constant 0 is called the zero polynomial. All algebraic expressions are not polynomials. An algebraic expression consists of a number of terms that are separated by '+' or '-' sign. These expressions may be simple or compound. One should note that 5/x,5 √x , 3+2/x² are all algebraic expressions but not polynomials. The powers of the variables in a polynomial are only whole numbers. 1/x is an algebraic expression but is not a polynomial.

A polynomial having only one term is called a monomial.-3x, 5.4 xy³z,7z/3 ,2x³y etc. are examples for monomials. The term 6/y is not a monomial because it is not a polynomial. A polynomial having two terms is called a binomial.6-5x, 4x³+6y, 3xyz+4 are all examples for binomials. A polynomial having 3 terms is called a trinomial. 5x+4y-2, 6y³+8y²-2y are examples for trinomials. In the polynomial, anxn +an-1xn-1 + an-2xn-2+............+ a1x + a0, if n ≠ 0 , then 'n' is the highest power of 'x' in the polynomial P(x) . So 'n' is called the degree of the polynomial. A polynomial of degree 1 is called a linear polynomial.2x+1, 3-z, 9.3y, y-4, x+2y+3z are examples for linear polynomial. The standard form of a linear polynomial in one variable is ax+b, where a,b are real numbers and a≠ 0. A polynomial of degree 2 is called a quadratic polynomial. 4-y², 5y², 9+3y+y², 8x-x²+11 are all examples for quadratic polynomial. The standard form of a quadratic polynomial in one variable is ax²+bx+c, where a,b,c are real numbers and a≠ 0 . A polynomial of degree 3 is called a cubic polynomial. 5x³, 2x³-7, 2z+6z²-7z³, y³+y²+y+5 are all examples for a cubic polynomial. The standard form of a cubic polynomial in one variable is ax³+bx²+cx+d, where a,b,c,d are real numbers and a ≠ 0. A polynomial of degree 'n' in one variable is of the form a_nxⁿ +a_n-1x^n-1 + a_n-2x^n-2+ ..................+ a₁x + a₀ , where a₀, a₁, ................a_n-1, a_n are real numbers and a_n≠ 0. In particular when a₁=a₂=................=a_n=0, we get a zero polynomial. The degree of the zero polynomial is not defined. The degree of non-zero constant polynomials is zero. There are also polynomials in more than one variable. x²+y²+xy, 3x²+8y²+4z-7, x³-z etc. are examples for polynomials in more than one variable.