Which of the following expressions are polynomials? Justify your answer:
(i) \(8\)
(ii) \(\sqrt{3}x^2 - 2x\)
(iii) \(1 - \sqrt{5}x\)
(iv) \(\dfrac{1}{5x^{-2}} + 5x + 7\)
(v) \(\dfrac{(x-2)(x-4)}{x}\)
(vi) \(\dfrac{1}{x+1}\)
(vii) \(\dfrac{1}{7}a^3 - \dfrac{2}{\sqrt{3}}a^2 + 4a - 7\)
(viii) \(\dfrac{1}{2x}\)
Polynomials: (i), (ii), (iv), (vii) because the exponent of the variable after simplification in each of these is a whole number.
Write whether the following statements are True or False. Justify your answer.
(i) A binomial can have at most two terms
(ii) Every polynomial is a binomial
(iii) A binomial may have degree 5
(iv) Zero of a polynomial is always 0
(v) A polynomial cannot have more than one zero
(vi) The degree of the sum of two polynomials each of degree 5 is always 5.
(i) False, because a binomial has exactly two terms.
(ii) False, \(x^3 + x + 1\) is a polynomial but not a binomial.
(iii) True, because a binomial is a polynomial whose degree is a whole number \(\ge 1\), so degree can be 5 also.
(iv) False, because zero of a polynomial can be any real number.
(v) False, a polynomial can have any number of zeroes. It depends upon the degree of the polynomial.
(vi) False, \(x^5 + 1\) and \(-x^5 + 2x + 3\) are two polynomials of degree 5 but the degree of the sum of the two polynomials is 1.