Q.1) In the following, circle the polynomials:
(i) [tex]\[\sqrt{x} + 2\][/tex]
(ii) [tex]\[t^2 + 5t - 1\][/tex]
(iii) [tex]\[\sqrt{3}x^2 - 2x\][/tex]
(iv) [tex]\[8\][/tex]
(v) [tex]\[1 - \sqrt{2}x\][/tex]
(vi) [tex]\[\frac{t}{4}\][/tex]
(vii) [tex]\[0\][/tex]
(viii) [tex]\[y^{\frac{3}{2}} + 3y + 2\][/tex]
Q.2) Mark as right (✔) or wrong (x):
1) [tex]\[x^2 + y + 3\][/tex] is a polynomial in one variable.
2) Number of terms in a polynomial is almost two.
3) 0 is a polynomial.
4) [tex]\[x + \frac{1}{x}\][/tex] is a polynomial.
Answer (1)
Let's break down the question and identify which expressions are polynomials and which statements about polynomials are correct.
Polynomials are algebraic expressions that consist of variables and coefficients, with non-negative integer exponents for the variables. Each separate part of a polynomial, separated by the '+' or '-' signs, is called a 'term'.
Q1: Identifying Polynomials
i) x + 2 : This is not a polynomial because the exponent of x is 2 1 , which is not a whole number.
(ii) t 2 + 5 t − 1 : This is a polynomial. It has whole number exponents (2, 1, 0) for the variable t .
(iii) 3 x 2 − 2 x : This is a polynomial because x has whole number exponents (2, 1) and 3 is a constant.
(iv) 8 : This constant is considered a polynomial. It can be thought of as 8 x 0 , where the exponent of x is 0, a whole number.
(v) 1 − 2 x : This is a polynomial. The variable x has a whole number exponent (1).
(vi) 4 t : This is a polynomial. It can be rewritten as 4 1 t 1 , with a whole number exponent.
(vii) 0 : This is a polynomial, known as the zero polynomial. It can be expressed as 0 x 0 .
(viii) y 2 3 + 3 y + 2 : This is not a polynomial because the exponent 2 3 is not a whole number.
Q2: Statements on Polynomials
x 2 + y + 3 is a polynomial in one variable. Incorrect (X). It contains two variables, x and y .
Number of terms in a polynomial is almost two. Incorrect (X). The number of terms in a polynomial can be any non-negative integer, not necessarily two.
0 is a polynomial. Correct (✔). It is the zero polynomial.
[ x + x 1 ] is a polynomial. Incorrect (X). The term x 1 has an exponent of − 1 , which is not a whole number.
