**Exercise 2.1 Page: 32**

**1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.**

**(i) 4x ^{2}–3x+7**

Solution:

The equation 4x^{2}–3x+7 can be written as 4x^{2}–3x^{1}+7x^{0}

Since *x* is the only variable in the given equation and the powers of x (i.e., 2, 1 and 0) are whole numbers, we can say that the expression 4x^{2}–3x+7 is a polynomial in one variable.

**(ii) y ^{2}+√2**

Solution:

The equation y^{2}+**√2** can be written as y^{2}+**√**2y^{0}

Since y is the only variable in the given equation and the powers of y (i.e., 2 and 0) are whole numbers, we can say that the expression y^{2}+**√**2 is a polynomial in one variable.

**(iii) 3√t+t√2**

Solution:

The equation 3√t+t√2 can be written as 3t^{1/2}+√2t

Though, *t* is the only variable in the given equation, the powers of *t* (i.e.,1/2) is not a whole number. Hence, we can say that the expression 3√t+t√2 is **not **a polynomial in one variable.

**(iv) y+2/y**

Solution:

The equation y+2/y an be written as y+2y^{-1}

Though, *y *is the only variable in the given equation, the powers of *y* (i.e.,-1) is not a whole number. Hence, we can say that the expression y+2/y is **not **a polynomial in one variable.

**(v) x ^{10}+y^{3}+t^{50}**

Solution:

Here, in the equation x^{10}+y^{3}+t^{50}

Though, the powers, 10, 3, 50, are whole numbers, there are 3 variables used in the expression

x^{10}+y^{3}+t^{50}. Hence, it is **not **a polynomial in one variable.

**2. Write the coefficients of x ^{2} in each of the following:**

**(i) 2+x ^{2}+x**

Solution:

The equation 2+x^{2}+x can be written as 2+(1)x^{2}+x

We know that, coefficient is the number which multiplies the variable.

Here, the number that multiplies the variable x^{2} is 1

, the coefficients of x^{2 }in 2+x^{2}+x is 1.

**(ii) 2–x ^{2}+x^{3}**

Solution:

The equation 2–x^{2}+x^{3 }can be written as 2+(–1)x^{2}+x^{3}

We know that, coefficient is the number (along with its sign, i.e., – or +) which multiplies the variable.

Here, the number that multiplies the variable x^{2} is -1

the coefficients of x^{2 }in 2–x^{2}+x^{3 }is -1.

**(iii) (/2)x ^{2}+x**

Solution:

The equation (/2)x^{2 }+x can be written as (/2)x^{2} + x

We know that, coefficient is the number (along with its sign, i.e., – or +) which multiplies the variable.

Here, the number that multiplies the variable x^{2} is /2.

the coefficients of x^{2 }in (/2)x^{2 }+x is /2.

**(iii)√2x-1**

Solution:

The equation √2x-1 can be written as 0x^{2}+√2x-1 [Since 0x^{2} is 0]

We know that, coefficient is the number (along with its sign, i.e., – or +) which multiplies the variable.

Here, the number that multiplies the variable x^{2}is 0

, the coefficients of x^{2 }in √2x-1 is 0.

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

Solution:

Binomial of degree 35: A polynomial having two terms and the highest degree 35 is called a binomial of degree 35

Eg., 3x^{35}+5

Monomial of degree 100: A polynomial having one term and the highest degree 100 is called a monomial of degree 100

Eg., 4x^{100}

**4. Write the degree of each of the following polynomials:**

**(i) 5x ^{3}+4x^{2}+7x**

Solution:

The highest power of the variable in a polynomial is the degree of the polynomial.

Here, 5x^{3}+4x^{2}+7x = 5x^{3}+4x^{2}+7x^{1}

The powers of the variable x are: 3, 2, 1

the degree of 5x^{3}+4x^{2}+7x is 3 as 3 is the highest power of x in the equation.

**(ii) 4–y ^{2}**

Solution:

The highest power of the variable in a polynomial is the degree of the polynomial.

Here, in 4–y^{2},

The power of the variable y is 2

the degree of 4–y^{2} is 2 as 2 is the highest power of y in the equation.

**(iii) 5t–√7**

Solution:

The highest power of the variable in a polynomial is the degree of the polynomial.

Here, in 5t**–√7 ,**

The power of the variable y is: 1

the degree of 5t**–√7 **is 1 as 1 is the highest power of y in the equation.

**(iv) 3**

Solution:

The highest power of the variable in a polynomial is the degree of the polynomial.

Here, 3 = 3×1 = 3× x^{0}

The power of the variable here is: 0

the degree of 3 is 0.

**5. Classify the following as linear, quadratic and cubic polynomials:**

Solution:

We know that,

Linear polynomial: A polynomial of degree one is called a linear polynomial.

Quadratic polynomial: A polynomial of degree two is called a quadratic polynomial.

Cubic polynomial: A polynomial of degree three is called a cubic polynomial.

**(i) x ^{2}+x**

Solution:

The highest power of x^{2}+x is 2

the degree is 2

Hence, x^{2}+x is a quadratic polynomial

**(ii) x–x ^{3}**

Solution:

The highest power of x–x^{3 }is 3

the degree is 3

Hence, x–x^{3} is a cubic polynomial

**(iii) y+y ^{2}+4**

Solution:

The highest power of y+y^{2}+4 is 2

the degree is 2

Hence, y+y^{2}+4is a quadratic polynomial

**(iv) 1+x**

Solution:

The highest power of 1+x is 1

the degree is 1

Hence, 1+x is a linear polynomial.

**(v) 3t**

Solution:

The highest power of 3t is 1

the degree is 1

Hence, 3t is a linear polynomial.

**(vi) r ^{2}**

Solution:

The highest power of r^{2 }is 2

the degree is 2

Hence, r^{2}is a quadratic polynomial.

**(vii) 7x ^{3}**

Solution:

The highest power of 7x^{3 }is 3

the degree is 3

Hence, 7x^{3} is a cubic polynomial.

Class 9th Maths Polynomial