NCERT Solution: Polynomials
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) 4x2 - 3x + 7
(ii) y2 + √2
(iii) 3√t + t√2
(iv) y + 2/y
(v) x10 + y3 + t50
Answer
(i) 4x2 - 3x + 7
There is only one variable x with whole number power so this polynomial in one variable.
(ii) y2 + √2
There is only one variable y with whole number power so this polynomial in one variable.
(iii) 3√2 + t√2
There is only one variable t but in 3√t power of t is 1/2 which is not a whole number so 3√t + t√2 is not a polynomial.
(iv) y + 2/y
There is only one variable y but 2/y = 2y-1 so the power is not a whole number so y + 2/y is not a polynomial.
(v) x10 + y3 + t50
There are three variable x, y and t and there powers are whole number so this polynomial in three variable.
Write the coefficients of x2 in each of the following:
(i) 2 + x2 + x
(ii) 2 - x2 + x3
(iv) √2x - 1
Answer
(i) coefficients of x2 = 1
(ii) coefficients of x2 = -1
(iii) coefficients of x2 = π/2
(iv) coefficients of x2 = 0
Degree of a polynomial is the highest power of the variable in the polynomial.
Binomial has two terms in it. Therefore, binomial of degree 35 can be written as x35+ x34.
Monomial has only one term in it. Therefore, monomial of degree 100 can be written as x100
Write the degree of each of the following polynomials:
(i) 5x3 + 4x2 + 7x
(ii) 4 – y2
(iii) 5t – √7
(iv) 3
Answer
(i) 5x3 has highest power in the given polynomial which power is 3. Therefore, degree of polynomial is 3.
(ii) – y2 has highest power in the given polynomial which power is 2. Therefore, degree of polynomial is 2.
(iii) 5t has highest power in the given polynomial which power is 1. Therefore, degree of polynomial is 1.
(iv) There is no variable in the given polynomial. Therefore, degree of polynomial is 0.
Classify the following as linear, quadratic and cubic polynomial:
(i) x2 + x
ANS: Quadratic Polynomial
(ii) x - x3
ANS: Cubic Polynomial
(iii) y + y2 +4
ANS: Quadratic Polynomial
(iv) 1 + x
ANS: Linear Polynomial
(v) 3t
ANS: Linear Polynomial
(vi) r2
ANS: Quadratic Polynomial
(vii) 7x3
ANS: Cubic Polynomial
Find the value of the polynomial at 5x + 4x2 + 3 at
(i) x = 0 (ii) x = - 1 (iii) x = 2
Answer
(i) p(x) = 5x + 4x2 + 3
p(0) = 5(0) + 4(0)2 + 3
= 3
(ii) p(x) = 5x + 4x2 + 3
p(-1) = 5(-1) + 4(-1)2 + 3
= 5 - 4(1) + 3 = -6
(iii) p(x) = 5x + 4x2 + 3
p(2) = 5(2) + 4(2)2 + 3
= 10 - 16 + 3 = -3
Find p(0), p(1) and p(2) for each of the following polynomials:
(i) p(y) = y2 - y + 1
(ii) p(t) = 2 + t + 2t2 - t3
(iii) p(x) = x3
(iv) p(x) = (x - 1) (x + 1)
Answer
(i) p(y) = y2 - y + 1
p(0) = (0)2 - (0) + 1 = 1
p(1) = (1)2 - (1) + 1 = 1
p(2) = (2)2 - (2) + 1 = 3
(ii) p(t) = 2 + t + 2t2 - t3
p(0) = 2 + 0 + 2 (0)2 - (0)3 = 2
p(1) = 2 + (1) + 2(1)2 - (1)3
= 2 + 1 + 2 - 1 = 4
p(2) = 2 + 2 + 2(2)2 - (2)3
= 2 + 2 + 8 - 8 = 4
(iii) p(x) = x3
p(0) = (0)3 = 0
p(1) = (1)3 = 1
p(2) = (2)3 = 8
(iv) p(x) = (x - 1) (x + 1)
p(0) = (0 - 1) (0 + 1) = (- 1) (1) = - 1
p(1) = (1 - 1) (1 + 1) = 0 (2) = 0
p(2) = (2 - 1 ) (2 + 1) = 1(3) = 3