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

**(i) 3x**^{2} – 4x + 15

^{2}– 4x + 15

**(ii) y**^{2 }+ 2√3

^{2 }+ 2√3

**(iii) 3√x + √2x**

**(iv) x – 4/x**

**(v) x**^{12} + y^{3 }+ t^{50}

^{12}+ y

^{3 }+ t

^{50}

**Solution:**

(i) 3x^{2}– 4x + 15It is a polynomial in one variable, that is, x. And all the powers of x are whole numbers.

(ii) y^{2}+ 2√3It is a polynomial in variable y. And all the powers of y are whole numbers.

(i

ii) 3√x + √2xIt is not a polynomial since the exponent of 3√x is a rational term.

(iv) x – 4/xIt is not a polynomial since the exponent of variable x is – 4/x which is not a positive term.

(v) x^{12}+ y^{3 }+ t^{50}It is a three variable polynomial, where the variables are x, y and t.

**Question 2: Write the coefficient of x**^{2} in each of the following:

^{2}in each of the following:

**(i) 17 – 2x + 7x**^{2}

^{2}

**(ii) 9 – 12x + x**^{3}

^{3}

**(iii) **π**/6 x**^{2} – 3x + 4

^{2}– 3x + 4

**(iv) √3x – 7**

**Solution:**

(i) 17 – 2x + 7x^{2}

Coefficient of x

^{2}in the above equation = 7

(ii) 9 – 12x + x^{3}Coefficient of x

^{2}=0, since there is no term with x^{2}

(iii)π/6 x^{2}– 3x + 4Coefficient of x2 in the above equation = π/6

(iv) √3x – 7Coefficient of x2 = 0, since there is no term with x2 in the above equation.

**Question 3: Write the degrees of each of the following polynomials:**

**(i) 7x**^{3} + 4x^{2} – 3x + 12

^{3}+ 4x

^{2}– 3x + 12

**(ii) 12 – x + 2x**^{3}

^{3}

**(iii) 5y – √2**

**(iv) 7**

**(v) 0**

**Solution:**

The degree is the highest possible degree of the variable in the polynomial. Now, we have

(i)Degree of the polynomial 7x^{3}+ 4x^{2}– 3x + 12 is 3, since the term x^{3}is highest

(ii)Degree of the polynomial 12 – x + 2x^{3}is 3, since the term x^{3 }is highest

(iii)Degree of the polynomial 5y – √2 is 1, since the term 5y only has the variable.

(iv)Degree of the polynomial 7 is 0, since there is no term with variable.

(v)Degree of the polynomial 0 is undefined.

**Question 4: Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:**

**(i) x + x**^{2} + 4

^{2}+ 4

**(ii) 3x – 2**

**(iii) 2x + x**^{2}

^{2}

**(iv) 3y**

**(v) t**^{2} + 1

^{2}+ 1

**(vi) 7t**^{4} + 4t^{3} + 3t – 2

^{4}+ 4t

^{3}+ 3t – 2

**Solution:**

Linear polynomials have highest degree = 1. Quadratic have have highest degree = 2. Cubic polynomials have have highest degree = 3 and bi-quadratic as 4.

(i)x + x^{2}+ 4: It is a quadratic polynomial since its highest possible degree is 2.

(ii)3x – 2 : It is a linear polynomial since its highest possible degree is 1.

(iii)2x + x^{2}: It is a quadratic polynomial since its highest possible degree is 2.

(iv)3y: It is a linear polynomial since its highest possible degree is 1.

(v)t^{2}+ 1: It is a quadratic polynomial since its highest possible degree s 2.

(vi)7t^{4}+ 4t^{3 }+ 3t – 2: It is a bi-quadratic polynomial since its highest possible degree is 4.