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