RD Sharma Solutions Class 9 Factorization Of Polynomials Exercise 6.1

RD Sharma Solutions Class 9 Chapter 6 Exercise 6.1

RD Sharma Class 9 Solutions Chapter 6 Ex 6.1 Free Download

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

        State the reasons for your answers

  1. \(3x^{2} – 4x + 15\)
  2. \(y^{2} + 2\sqrt{3}\)
  3. \(3\sqrt{x} + \sqrt{2}x\)
  4. \(x – \frac{4}{x}\)
  5. \(x^{12} + y^{2} + t^{50}\)

Sol :

  1. \(3x^{2} – 4x + 15\) – it is a polynomial of x
  2. \(y^{2} + 2\sqrt{3}\) – it is a polynomial of y
  3. \(3\sqrt{x} + \sqrt{2}x\) – it is not a polynomial since the exponent of \(3\sqrt{x}\) is not a positive term
  4. \(x-\frac{4}{x}\)– it is not a polynomial since the exponent of – \(\frac{4}{x}\) is not a positive term
  5. \(x^{12} + y^{2} + t^{50}\) – it is a three variable polynomial which variables of x, y, t

Q2. Write the coefficients of \(x^{2}\) in each of the following

  1. \(17 – 2x + 7x^{2}\)
  2. \(9 – 12x + x^{2}\)
  3. \(\frac{\prod }{6}x^{2} – 3x + 4\)
  4. \(\sqrt{3}x – 7\)

Sol :

Given , to find the coefficients of  \(x^{2}\)

  1. \(17 – 2x + 7x^{2}\) – the coefficient is 7
  2. \(9 – 12x + x^{2}\) – the coefficient is 0
  3. \(\frac{\prod }{6}x^{2} – 3x + 4\) – the coefficient is \(\frac{\prod }{6}\)
  4. \(\sqrt{3}x – 7\) – the coefficient is 0

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

  1. \(7x^{3} + 4x^{2} – 3x + 12\)
  2. \(12 – x + 2x^{2}\)
  3. \(5y – \sqrt{2}\)
  4. \(7- 7x^{0}\)
  5. 0

Sol :

Given , to find degrees of the polynomials

Degree is highest power in the polynomial

  1. \(7x^{3} + 4x^{2} – 3x + 12\) – the degree is 3
  2. \(12 – x + 2x^{3}\) – the degree is 3
  3. \(5y – \sqrt{2}\) – the degree is 1
  4. \(7- 7x^{0}\) – 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. \(x + x^{2} + 4\)
  2. 3x – 2
  3. \(2x + x^{2}\)
  4. 3y
  5. \(t^{2} + 1\)

f . \(7t^{4} + 4t^{2} + 3t – 2\)

Sol :

Given

  1. \(x + x^{2} + 4\) – it is a quadratic polynomial as its degree is 2
  2. 3x – 2 – it is a linear polynomial as its degree is 1
  3. \(2x + x^{2}\) – it is a quadratic polynomial as its degree is 2
  4. 3y – it is a linear polynomial as its degree is 1
  5. \(t^{2} + 1\) – it is a quadratic polynomial as its degree is 2

f . \(7t^{4} + 4t^{2} + 3t – 2\) – 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. \(x^{2} – xy + 7y^{2}\)
  2. \(x^{2} – 2tx + 7t^{2} – x + t\)
  3. \(t^{3} – 3t^{2} + 4t – 5\)
  4. xy + yz + zx

Sol :

Given

  1. \(x^{2} – xy + 7y^{2}\) – it is a polynomial in two variables x and y
  2. \(x^{2} – 2tx + 7t^{2} – x + t\) – it is a polynomial in two variables x and t
  3. \(t^{3} – 3t^{2} + 4t – 5 \)– 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 :

  1. \(f(x) = 4x^{3} – x^{2} -3x + 7\)
  2.   b . \(g(x) = 2x^{3} – 3x^{2} + \sqrt{x} – 1\)
  3. \(p(x) = \frac{2}{3}x^{2} + \frac{7}{4}x + 9\)
  4. \(q(x) = 2x^{2} – 3x + \frac{4}{x} + 2\)
  5. \(h(x) = x^{4} – x^{\frac{3}{2}} + x – 1\)
  6. \(f(x) = 2 + \frac{3}{x} + 4x\)

Sol :

Given

  1. \(f(x) = \(4x^{3} – x^{2} -3x + 7\)\(4x^{3} – x^{2} -3x + 7\]”> – it is a polynomial
  2.   b . \(g(x) = 2x^{3} – 3x^{2} + \sqrt{x} – 1\) – it is not a polynomial since the exponent of  \(\sqrt{x}\) is a negative integer
  3. \(p(x) = \(\frac{2}{3}x^{2} + \frac{7}{4}x + 9\)\(\frac{2}{3}x^{2} + \frac{7}{4}x + 9\]”> – it is a polynomial as it has positive integers as exponents
  4. \(q(x) = 2x^{2} – 3x + \frac{4}{x} + 2\) – it is not a polynomial since the exponent of  \(\frac{4}{x}\)  is a negative integer
  5. h(x) = \(x^{4} – x^{\frac{3}{2}} + x – 1\) – it is not a polynomial since the exponent of – \(x^{\frac{3}{2}}\)   is a negative integer
  6. f(x) = 2 + \(\frac{3}{x} + 4x\) – it is not a polynomial since the exponent of  \(\frac{3}{x}\)   is a negative integer

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

  1. \(f(x) = 0\)
  2. \(g(x) = 2x^{3} – 7x + 4\)
  3. \(h(x) = -3x + \frac{1}{2}\)
  4. \(p(x) = 2x^{2} – x + 4\)
  5. \(q(x) = 4x + 3\)
  6. \(r(x) = 3x^{3} + 4x^{2} + 5x – 7\)

Sol :

Given ,

  1. \(f(x) = 0\) – as 0 is constant , it is a constant variable
  2. \(g(x) = 2x^{3} – 7x + 4\) – since the degree is 3 , it is a cubic polynomial
  3. \(h(x) = -3x + \frac{1}{2}\) – since the degree is 1 , it is a linear polynomial
  4. \(p(x) = 2x^{2} – x + 4\) – 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. \(r(x) = 3x^{3} + 4x^{2} + 5x – 7\) – 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 – \(2t^{100}\)


Practise This Question

The relation R is defined on the set of natural numbers as {(a,b) : a = 2b}. Then R1 is given by