Class 9 RD Sharma Solutions – Chapter 6 Factorisation of Polynomials- Exercise 6.1

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

(i) 3x² – 4x + 15

(ii) y² + 2√3

(iii) 3√x + √2x

(iv) x – 4/x

(v) x¹² + y³ + t⁵⁰

Solution:

(i) 3x² – 4x + 15

It is a polynomial in one variable, that is, x. And all the powers of x are whole numbers.

(ii) y² + 2√3

It is a polynomial in variable y. And all the powers of y are whole numbers.

(iii) 3√x + √2x

It is not a polynomial since the exponent of 3√x is a rational term.

(iv) x – 4/x

It is not a polynomial since the exponent of variable x is – 4/x which is not a positive term.

(v) x¹² + y³ + t⁵⁰

It is a three variable polynomial, where the variables are x, y and t.

Question 2: Write the coefficient of x² in each of the following:

(i) 17 – 2x + 7x²

(ii) 9 – 12x + x³

(iii) π/6 x² – 3x + 4

(iv) √3x – 7

Solution:

(i) 17 – 2x + 7x²

Coefficient of x² in the above equation = 7

(ii) 9 – 12x + x³

Coefficient of x² =0, since there is no term with x²

(iii) π/6 x² – 3x + 4

Coefficient of x2 in the above equation = π/6

(iv) √3x – 7

Coefficient 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) 7x³ + 4x² – 3x + 12

(ii) 12 – x + 2x³

(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 7x³ + 4x² – 3x + 12 is 3, since the term x³ is highest

(ii) Degree of the polynomial 12 – x + 2x³ is 3, since the term x³ 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 + x² + 4

(ii) 3x – 2

(iii) 2x + x²

(iv) 3y

(v) t² + 1

(vi) 7t⁴ + 4t³ + 3t – 2

Solution:

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

(i) x + x² + 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 + x²: 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) t²+ 1: It is a quadratic polynomial since its highest possible degree s 2.

(vi) 7t⁴ + 4t³ + 3t – 2: It is a bi-quadratic polynomial since its highest possible degree is 4.