Thursday, July 29, 2010

Exercise 2.1


Question 1:
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i)  (ii)  (iii)
(iv)  (v) 

Ans.
(i)
Yes, this expression is a polynomial in one variable x.
(ii)
Yes, this expression is a polynomial in one variable y.
(iii)
No. It can be observed that the exponent of variable t in term  is , which is not a whole number. Therefore, this expression is not a polynomial.
(iv)
No. It can be observed that the exponent of variable y in termis −1, which is not a whole number. Therefore, this expression is not a polynomial.
(v)
No. It can be observed that this expression is a polynomial in 3 variables xy, and t. Therefore, it is not a polynomial in one variable.

Question 2:
Write the coefficients of in each of the following:
(i)  (ii) 
(iii)  (iv) 

Ans.
(i)
In the above expression, the coefficient of is 1.
(ii)
In the above expression, the coefficient of is −1.
(iii)
In the above expression, the coefficient of  is.
(iv)
In the above expression, the coefficient of  is 0.

Question 3:
Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Ans.
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 .
Monomial has only one term in it. Therefore, monomial of degree 100 can be written as x100.

Question 4:
Write the degree of each of the following polynomials:
(i) (ii)
(iii) (iv) 3

Ans.
Degree of a polynomial is the highest power of the variable in the polynomial.
(i)
This is a polynomial in variable x and the highest power of variable x is 3. Therefore, the degree of this polynomial is 3.
(ii)
This is a polynomial in variable y and the highest power of variable y is 2. Therefore, the degree of this polynomial is 2.
(iii) 
This is a polynomial in variable t and the highest power of variable t is 1. Therefore, the degree of this polynomial is 1.
(iv) 3
This is a constant polynomial. Degree of a constant polynomial is always 0.




Question 5:
Classify the following as linear, quadratic and cubic polynomial:
(i) (ii)  (iii)  (iv)  (v) 
(vi)  (vii) 

Ans.
Linear polynomial, quadratic polynomial, and cubic polynomial has its degrees as 1, 2, and 3 respectively.
(i)  is a quadratic polynomial as its degree is 2.
(ii) is a cubic polynomial as its degree is 3.
(iii)  is a quadratic polynomial as its degree is 2.
(iv) 1 + x is a linear polynomial as its degree is 1.
(v)  is a linear polynomial as its degree is 1.
(vi) is a quadratic polynomial as its degree is 2.
(vii) is a cubic polynomial as its degree is 3.