RD Sharma 2020 solution class 9 chapter 6 Factorization of polynomial Expressions Exercise 6.1

Exercise 6.1

Page-6.2

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$\sqrt{3}$

(iii) $3\sqrt{x}+\sqrt{2}x$

(iv) $x-\frac{4}{x}$

(v) x¹² + y³ + t⁵⁰

Answer 1:

(i) is a polynomial of degree 2.i.e Quadratic polynomial.

(ii) is a polynomial of degree 2 in y variable. i.e. Quadratic polynomial.

(iii)

It is not a polynomial because exponent of x is 1/2 which is not a positive integer.

(iv)

It is not a polynomial because is fractional part.

(v)

It is a polynomial in three variables 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) $\frac{\mathrm{\pi }}{6}{x}^2-3x+4$

(iv) $\sqrt{3}x-7$

Answer 2:

(i)

Coefficient of

(ii)

Coefficient of

(iii)

Coefficient of

(iv)

Coefficient of

Page-6.3

Question 3:

Write the degrees of each of the following polynomials:

(i) 7x³ + 4x² − 3x + 12

(ii) 12 − x + 2x³

(iii) $5y-\sqrt{2}$

(iv) 7

(v) 0

Answer 3:

(i) 7x³ + 4x² − 3x + 12

Degree of the polynomial = 3

Because the highest power of x is 3.

(ii)

Degree of the polynomial = 3. Because the highest power of x is 3.

(iii)

Degree of the polynomial = 1. Because the highest power of y is 1.

(iv) 7

Degree of the polynomial = 0. Because there is no variable term in the expression

(v) 0

Degree of the polynomial is not defined. As there is no variable or constant term

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

Answer 4:

(i)

The degree of the polynomial is 2. It is quadratic in x.

So, it is quadratic polynomial.

(ii)

The degree of the polynomial is 1. It is a linear polynomial in x.

(iii)

The degree of the polynomial is 1.

It is linear a polynomial in x.

(iv) 3y

The degree of the polynomial is 1. It is linear in y.

(v)

The degree of the polynomial is 2. It is quadratic polynomial in t.

(vi)

The degree of the polynomial is 4. Therefore, it is bi-quadratic polynomial in t.

Question 5:

Classify the following polynomials as polynomials in one-variable, two variables etc.:

(i) x² − xy + 7y²

(ii) x² − 2tx + 7t² − x + t

(iii) t³ − 3t² + 4t − 5

(iv) xy + yz + zx

Answer 5:

(i)

Here, x and y are two variables.

So, it is polynomial in two variables.

(ii)

Here, x and t are two variables.

So, it is polynomial in two variables.

(iii)

Here, only t is variable.

So, it is polynomial in one variable.

(iv)

Here, x, y and z are three variables

So, it is polynomial in three variables.

Question 6:

Identify polynomials in the following:

(i) f(x) = 4x³ − x² − 3x + 7

(ii) g(x) = 2x³ − 3x² + $\sqrt{x}$ − 1

(iii) p(x) = $\frac{2}{3}{x}^2-\frac{7}{4}x+9$

(iv) q(x) = 2x² − 3x + $\frac{4}{x}$+ 2

(v) h(x) = $x^4-x^{\frac{3}{2}}+x-1$

(vi) f(x) = $2 +\frac{3}{x}+4x$

Answer 6:

(i)

It is cubic in x, so, it is cubic polynomial in x variable.

 

(ii)

Here, exponent of x in is not a positive integer, so, it is not a polynomial.

 

(iii)

It is a quadratic polynomial.

 

(iv) q(x) = 2x² − 3x + $\frac{4}{x}$+ 2

Here, exponent of x in is not a positive integer. So it is not a polynomial.

 

(v)

Here, exponent of x in x^3/2 is not a positive integer. So, it is not a polynomial.

 

(vi)

Here, exponent of x in is not a positive integer, so, it not a polynomial.

 

Question 7:

Identify constant, linear, quadratic and cubic polynomials from the following polynomials:

(i) f(x) = 0

(ii) g(x) = 2x³ − 7x + 4

(iii) h(x) = -$3x+\frac{1}{2}$

(iv) p(x) = 2x² − x + 4

(v) q(x) = 4x + 3

(vi) r(x) = 3x³ + 4x² + 5x − 7

Answer 7:

(i)

The given expression is a Constant polynomial as there is no variable term in it.

 

(ii)

The given expression is Cubic polynomial as the highest exponent of x is 3.

 

(iii)

The given expression is linear polynomial as the highest exponent of x is 1.

 

(iv)

The given expression is Quadratic polynomial as the highest exponent of x is 2.

 

(v)

The given polynomial is an linear polynomial as the highest exponent of x is 1.

 

(vi)

The given polynomial is Cubic polynomial as the highest exponent of x is 3.

Question 8:

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

Answer 8:

An example of binomial of degree 35 is $f\left(t\right) = 4t^{35}-\frac{1}{2}$. It is a binomial as it has two terms and degree is 35 because highest exponent of t is 35.

An example of monomial of degree 100 is . It is a monomial as it has only one term and degree is 100 because highest exponent of x is 100