RD Sharma Solutions Ex-6.1, Factorization Of Polynomials, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics PDF Download

Q1. Which of the following expressions are polynomials in one variable and which are not?

1. 3x²–4x+15
2. y²+2√3
3. 3√x+√2x
4.
5. x¹²+y²+t⁵⁰

Sol :

1. 3x²–4x+15 – it is a polynomial of x

2. y²+2√3 – it is a polynomial of y

3. 3√x+√2x – it is not a polynomial since the exponent of 3√x is not a positive term

4.   it is not a polynomial since the exponent of  is not a positive term

5. x¹²+y²+t⁵⁰ – it is a three variable polynomial which variables of x, y, t

Q2. Write the coefficients of x² in each of the following

Sol :

Given , to find the coefficients of x²

1. 17–2x+7x² – the coefficient is 7

2. 9–12x+x² – the coefficient is 0

3.   the coefficient is

4.  the coefficient is 0

Q3. Write the degrees of each of the following polynomials :

1. 7x³+4x²–3x+12
2. 12–x+2x²
3. 5y–√2
4. 7−7x⁰
5. 0

Sol :

Given , to find degrees of the polynomials

Degree is highest power in the polynomial

1. 7x3+4x²–3x+12 – the degree is 3

2. 12–x+2x³ – the degree is 3

3. 5y–√2 – the degree is 1

4. 7−7x⁰ – the degree is 0

5. 0 – the degree of 0 is not defined

Q4. Classify the following polynomials as linear, quadratic, cuboc and biquadratic polynomials :

1. x+x²+4
2. 3x – 2
3. 2x+x²
4. 3y
5. t²+1

f . 7t⁴+4t²+3t–2

Sol :

Given

1. x+x²+4 – it is a quadratic polynomial as its degree is 2

2. 3x – 2 – it is a linear polynomial as its degree is 1

3. 2x+x² – it is a quadratic polynomial as its degree is 2

4. 3y – it is a linear polynomial as its degree is 1

5. t²+1 – it is a quadratic polynomial as its degree is 2

f . 7t⁴+4t²+3t–2 – it is a bi- quadratic polynomial as its degree is 4

Q5. Classify the following polynomials as polynomials in one variables, two – variables etc :

1. x²–xy+7y²
2. x²–2tx+7t²–x+t
3. t³–3t²+4t–5
4. xy + yz + zx

Sol :

Given

1. x²–xy+7y² – it is a polynomial in two variables x and y

2. x²–2tx+7t²–x+t – it is a polynomial in two variables x and t

3. t³–3t²+4t–5– it is a polynomial in one variable t

4. xy+yz+zx – it is a polynomial in 3 variables in x , y and z

Q6. Identify the polynomials in the following :

Sol :

Given

1. f(x) = 4x³–x²−3x+7 – it is a polynomial

2. b. g(x) = 2x³–3x²+√x–1 – it is not a polynomial since the exponent of √x is a negative integer

3.   it is a polynomial as it has positive integers as exponents 

4.   it is not a polynomial since the exponent of 4/x is a negative integer

5.  it is not a polynomial since the exponent of   is a negative integer 

6.   it is not a polynomial since the exponent of 3/x is a negative integer

Q7. Identify constant , linear , quadratic abd cubic polynomial from the following polynomials :

1. f(x) = 0
2. g(x) = 2x³–7x+4
3.
4. p(x) = 2x²–x+4
5. q(x) = 4x+3
6. r(x) = 3x³+4x²+5x–7

Sol :

Given ,

1. f(x) = 0 – as 0 is constant , it is a constant variable

2. g(x) = 2x³–7x+4 – since the degree is 3 , it is a cubic polynomial

3.  – since the degree is 1 , it is a linear polynomial

4. p(x) = 2x²–x+4 – since the degree is 2 , it is a quadratic polynomial

5. q(x) = 4x+3 – since the degree is 1 , it is a linear polynomial

6. r(x) = 3x³+4x²+5x–7 – since the degree is 3 , it is a cubic polynomial

Q8. Give one example each of a binomial of degree 25, and of a monomial of degree 100

Sol :

Given , to write the examples for binomial and monomial with the given degrees

Example of a binomial with degree 25 – 7x^35–5

Example of a monomial with degree 100 – 2t¹⁰⁰