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Introduction of PolynomialsPolynomials are algebraic expressions that include real numbers and variables. Division and square roots cannot be involved in the variables. The variables can only include addition, subtraction and multiplication.Let x be a variable, n be a positive integer and a _{0} , a _{1} ,a _{2} ... ,a _{n} be constants. Then, f(x) = a _{n} x ^{n} + a _{n1} x ^{n1} + ... + a _{1} x + a _{0} is called a Polynomial in variable x. Here, a _{n} x ^{n} , a _{n1} x ^{n1} , ... , a _{1} x, a _{0} are known as terms and a _{n} , a _{n1} , ..., a _{1} ,a _{0} are coefficients . Examples 1) p(x) = 3x 2 > polynomial in variable x. 2) q(y) = 3y ^{2}  2y + 4 > is a polynomial in y. 3) f(u)= 1/2 u ^{3}  3u ^{2} + 2u  4 > is a polynomial in variable u. Not a Polynomial : If the exponent of any term is less than 1 or negative. Examples : 1) 2x ^{2}  3√x + 5 Solution : It is not a polynomial because the exponent of one term is 1/2 which is less than 1. 2) 1/(x ^{2}  2x + 5) Solution : It is not a polynomial because the exponent is negative. 3) 2x ^{3}  3/x + 4 Solution : As the exponent of 2nd term is 1 so it is not a polynomial. _________________________________________________________________ Q.1 Write any polynomial in x with degree 3. Q.2 P(x) = x ^{3} + 2 ^{3}  4x + 8. Write the coefficient of x ^{2} . Q.3 In P(x) = 5x ^{2} + x ^{5}  2x + 4 write the degree of polynomial. Q.4 In P(x) = 1/2 x ^{3}  7x ^{2} + 8x, state the number of terms and write each terms. Polynomials • Degree of the Polynomial • Zeros of Polynomial • Remainder Theorem • Find remainder by Synthetic Division • Rational root test in Polynomial • Solved Examples on Factorization
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