What are the two conditions for which the expression p ( x ) = a n x n + a n - 1 x n - 1 + ⋯ ⋯ ⋯ + a 1 x + a 0 is a polynomial ?

If x be a variable, n be a whole number and a 0 , a 1 , a 2 , … a n be all constant real numbers, then the expression. a n x n + a n - 1 x n - 1 + a n - 2 x n - 2 + ⋯ ⋯ + a 1 x + a 0 is called a polynomial of the variable x . Here, the two conditions for the expression is a polynomial is (i) the powers of the polynomials should not be negative. (ii) Polynomials in two variables are algebraic expressions consisting of terms in the form a x n y m . The degree of each term in a polynomial in two variables is the sum of the exponents in each term and the degree of the polynomial is the largest such sum.

Is 11 3 a polynomial? Give reasons in favour of your answer.

If x be a variable, n be a whole number and a 0 , a 1 , a 2 , … a n a n ≠ 0 be all constant real numbers, then the expression, a n x n + a n - 1 x n - 1 + a n - 2 x n - 2 + ⋯ ⋯ + a 1 x + a 0 is called a polynomial of the variable x . Given expression is 11 3 , The above expression is not a polynomial, because the indices of the term is not a whole number. i.e, the index of 11 3 = 11 1 3 is 1 3 is not a whole number. Hence, the given expression is not a polynomial.

If m be a whole number, then determine the degree of the polynomial x m + x m - 1 + x m - 2 + ⋯ ⋯ + x + 1 . Also, find the co-efficient of x m - 3 of the polynomial.

Given polynomial is x m + x m - 1 + x m - 2 + ⋯ ⋯ + x + 1 , where m is a whole number. We know that, the degree of a polynomial is the highest power in the expression. So, the degree of the polynomial is m . The polynomial can be expanded as x m + x m - 1 + x m - 2 + x m - 3 + x m - 4 ⋯ ⋯ + x + 1 We can see that the coefficient of x m - 3 is 1 as the coefficient is the constant multiplied to the variable. Hence, the degree of the polynomial is m and the coefficient of x m - 3 is 1 .

Determine the degree and number of terms of the polynomial x 3 - 1 1 + x + x 2 k + 2 where k = 8 .

Given polynomial is x 3 - 1 1 + x + x 2 k + 2 x 3 - 1 1 + x + x 2 k + 2 = ( x - 1 ) x 2 + x + 1 x 2 + x + 1 k + 2 = ( x - 1 ) k + 2 Degree of the above polynomial is k + 2 = 8 + 2 = 10 . (Given k = 8 ) We know that the number of terms of the polynomial is one more than the degree of the polynomial. Hence, the number of terms of the polynomial is 10 + 1 = 11 .