Class 9 RD Sharma Solutions Chapter 6 Factorisation of Polynomials Exercise 6.1 (original) (raw)
Last Updated : 23 Jul, 2025
Factorisation is a critical algebraic technique used to simplify polynomials and solve equations. In Class 9, RD Sharma's textbook introduces students to the fundamentals of factorization in Chapter 6. Exercise 6.1 focuses on applying these concepts to factorize polynomials effectively which is foundational for mastering higher algebraic topics.
Factorisation of Polynomials
The Factorisation of polynomials involves breaking down a polynomial into the product of the simpler polynomials or factors. This process is crucial for solving polynomial equations and simplifying expressions. The main techniques include factoring out the greatest common divisor (GCD) using special formulas and applying algebraic identities. By factoring polynomials, we can solve equations simplify expressions, and understand polynomial behavior more deeply.
**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
****(ii) y** 2 **+ 2√3
****(iii) 3√x + √2x**
****(iv) x – 4/x**
****(v) x** 12 + y 3 **+ t 50
**Solution:
****(i) 3x** 2 – 4x + 15
It is a polynomial in one variable, that is, x. And all the powers of x are whole numbers.
****(ii) y** 2 + 2√3
It is a polynomial in variable y. And all the powers of y are whole numbers.
(i**ii) 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) 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:
****(i) 17 – 2x + 7x** 2
****(ii) 9 – 12x + x** 3
(iii) π/6 x 2 – 3x + 4
****(iv) √3x – 7**
**Solution:
****(i) 17 – 2x + 7x** 2
Coefficient of x2 in the above equation = 7
****(ii) 9 – 12x + x** 3
Coefficient of x2 =0, since there is no term with x2
****(iii) π/6 x** 2 – 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) 7x** 3 + 4x 2 – 3x + 12
****(ii) 12 – x + 2x** 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 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 + x** 2 + 4
****(ii) 3x – 2**
****(iii) 2x + x** 2
****(iv) 3y**
****(v) t** 2 + 1
****(vi) 7t** 4 + 4t 3 + 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.
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Conclusion
The Factorisation of polynomials is a vital skill in algebra simplifying the complex expressions and solving polynomial equations. Exercise 6.1 in RD Sharma's Class 9 textbook introduces key factorisation techniques that are foundational for the mastering more advanced algebraic concepts. By practicing these techniques, students build a strong foundation in the algebra preparing them for the future mathematical challenges.