# RD Sharma Solutions Class 9 Factorization Of Polynomials Exercise 6.1

## RD Sharma Solutions Class 9 Chapter 6 Ex 6.1

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

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 – $2t^{100}$

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