Algebra Polynomial(Lecture-I) Dr. Amal Kumar Adak Department of Mathematics, G.D.College, Begusarai March 27, 2020 Algebra Dr. Amal Kumar Adak Definition Function: Any mathematical expression involving a quantity (say x) which can take different values, is called a function of that quantity. The quantity (x) is called the variable and the function of it is denoted by f (x) ; F (x) etc. Example:f (x) = x2 + 5x + 6 + x6 + x3 + 4x5. Algebra Dr. Amal Kumar Adak Polynomial Definition Polynomial: A polynomial in x over a real or complex field is an expression of the form n n−1 n−3 p (x) = a0x + a1x + a2x + ::: + an,where a0; a1;a2; :::; an are constants (free from x) may be real or complex and n is a positive integer. Example: (i)5x4 + 4x3 + 6x + 1,is a polynomial where a0 = 5; a1 = 4; a2 = 0; a3 = 6; a4 = 1a0 = 4; a1 = 4; 2 1 3 3 2 1 (ii)x + x + x + 2 is not a polynomial, since 3 ; 3 are not integers. (iii)x4 + x3 cos (x) + x2 + x + 1 is not a polynomial for the presence of the term cos (x) Algebra Dr. Amal Kumar Adak Definition Zero Polynomial: A polynomial in which all the coefficients a0; a1; a2; :::; an are zero is called a zero polynomial. Definition Complete and Incomplete polynomials: A polynomial with non-zero coefficients is called a complete polynomial. Other-wise it is incomplete. Example of a complete polynomial:x5 + 2x4 + 3x3 + 7x2 + 9x + 1. Example of an incomplete polynomial: x5 + 3x3 + 7x2 + 9x + 1. Algebra Dr. Amal Kumar Adak Definition Degree of a polynomial: The degree of p (x) is the largest non-negative integer i such that the coefficient of xi in p (x) is non zero. Example : (i) If a0 6= 0 then the degree of the polynomial p (x) is n. (ii) The degree of the polynomial 4x4 + 5x3 + 6x + 1 is 4. (iii) The degree of the polynomial x5 + x3 + x2 + x + 1 is 5. (iv) The degree of a zero polynomial is defined to be −∞. Algebra Dr. Amal Kumar Adak Definition Binomial: A polynomial which contains exactly two terms is called a binomial. For example a0x + a1 is a binomial. Similarly a polynomial which contains exactly one term is called a monomial( for example 3x) and a polynomial which contains three terms is called trinomial (for example 3x2 + 6x + 2). Polynomial of more than one variable: Example:5x5 + 6x3y 4 + 2x2y 3 + 6y 5 Algebra Dr. Amal Kumar Adak Definition Homogeneous polynomial: A polynomial is called a homogeneous polynomial if the sum of the powers of each term is equal. Example : 2x7 + 5x5y 2 + 7x6y + 3x4y 3 + xy 6 + y 7 is a homogeneous polynomial of degree 7 and sum of powers of each term is 7. Algebra Dr. Amal Kumar Adak Properties of polynomials: (i)Equality: Two polynomials of same n n−1 degreef (x) = a0x + a1x + ::: + anand n n−1 g (x) = b0x + b1x + ::: + bn are said to be equal if a0 = b0; a1 = b1; :::; an = bn, i.e. the coefficients of like powers are equal. (ii)Addition: The addition of two polynomials of same degree n n−1 f (x) = a0x + a1x + ::: + an and n n−1 g (x) = b0x + b1x + ::: + bn is given by n n−1 f (x) + g (x) = (a0 + b0) x + (a1 + b1) x + ::: + (an + bn). Algebra Dr. Amal Kumar Adak The addition of two polynomials of different degree n n−1 f (x) = a0x + a1x + ::: + an and m m−1 g (x) = b0x + b1x + ::: + bm is given by f (x) + g (x) n m = a0x + ··· + (am + b0) x m−1 + (am−1 + b1) x + ::: + (an + bm) ifn > m. f (x) + g (x) = m n n−1 b0x + ::: + (bn + a0) x + (bn−1 + a1) x + ::: + (an + bm) ifm > n Ex: 1. Let f (x) = 2x2 + 3x + 5 and g (x) = 5x2 + 6x + 1. Then f (x) + g (x) = 7x2 + 9x + 6. 2. Let f (x) = 12x5 + 9x4 + 3x2 + 2x + 7 and g (x) = 3x4 + 8 .Then f (x) + g (x) = 12x5 + 12x4 + 3x2 + 2x + 15 Algebra Dr. Amal Kumar Adak n n−1 n−2 If f (x) = a0x + a1x + a2x + ::: + an and m m−1 m−2 g (x) = b0x + b1x + b2x + ::: + bm be two polynomials of degree m and n respectively then there product f (x) :g (x) is given Pm+n m+n−i Pn by f (x) :g (x) = i=0 ci x where ci = j=0 ai :bj for i = 0; 1; 2; :::; n. Ex: Let f (x) = 3x2 + 2x + 4 and g (x) = 5x + 6 then f (x) :g (x) = 15x3 + 28x2 + 32x + 24 Algebra Dr. Amal Kumar Adak Let f (x)and g (x)be two polynomials of degree n and m respectively where n > m then there exists two polynomials q (x) and r (x) such that f (x) = g (x) :q (x) + r (x) where q (x) is a polynomial of degree n − m is called quotient and r (x)is another polynomial either zero or degree less than m Ex: If f (x) = x2 + 4x + 2 and g (x) = x + 2 then f (x) = g (x) :q (x) + R, where q (x) = x + 2 and R = 2 Algebra Dr. Amal Kumar Adak.