**Polynomials**

**Question 1.****Which of the following expressions are polynomials in one variable and which are not? State reason for your answer?**

- 4x
^{2}-3x + 7 - y
^{2}+ - 3 + t
- y +
- x
^{10}+ y^{3}+t^{50}

**Solution:**

(1) 4x^{2}– 3x +7 is an expression having only non-negative integral powers of x. So, it is a polynomial.

(2) y^{2} + is an expression having only non-negative integral powers of So, it is a polynomial.

(3) 3 + t is an expression in which one term namely 3 has rational power to t. So, it is not a polynomial.

(4) y + is an expression in which one term namely y +

=> i.e., 2y^{-3} has negative power of y. So, it is not a polynomial.

(5) x^{10} + y^{3} +t^{50} is an expression which has 3 variables.

**Question 2.****Write the coefficients of x ^{2} in each of the following:**(i) 2 + x

^{2}+x

(ii) 2-x

^{2}+x

^{a}(iii)

(iv) x -1

**Solution:**

(i) The coefficient of x^{2} in 2 + x^{2} + x is 1.

(ii) The coefficient of x^{2} in 2 – x^{2} + x^{3} is -1.

(iii) The coefficient of x^{2} in + x is .

(iv) The coefficient of x^{2} in x -1 is 0.

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

**Solution:**

(1) y^{35} + 2 is a binomial of degree 35.

(2) y^{100} is a monomial of degree 100.

**Question 4.****Write the degree of each of the following polynomials :**(i) 5x

^{3}+ 4x

^{2}+ 7x

(ii) 4 – y

^{2}(iii) 5t-

(iv) 3

**Solution:**

(i) The highest power term is 5x^{3} and the exponent is 3. So, the degree is 3.

(ii) The highest power term is -y^{2} and the exponent is 2. So, the degree is 2.

(iii) The highest power term is 5t and the exponent is 1. So, the degree is 1.

(iv) The only term here is 3 which can be written as 3x° and so the exponent is 0.

Therefore, the degree is 0.

**Question 5.****Classify the following as linear, quadratic and cubic polynomials**:

(i) x^{2}+x

(ii) x^{2}-x

(iii) y + y^{2}+4

(iv) 1 + x

(v) 3t

(vi) r^{2}(vii) 7r^{3}

**Solution:**

(i) The highest degree of x^{2} + x is 2, so it is a quadratic polynomial.

(ii) The highest degree of x – x^{3} is 3, so it is a cubic polynomial.

(iii) The highest degree of y + y^{2} + 4 is 2, so it is a quadratic polynomial.

(iv) The highest degree of x in 1 + x is 1, so it is a linear polynomial.

(v) The highest degree oft in 3t is 1, so it is a linear polynomial.

(vi) The highest degree of r in r^{2} is 2, so it is a quadratic polynomial.

(vii) The highest degree of x in 7x^{3} is 3, so, it is a cubic polynomial.