## 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) 4x2–3x+7**

Solution:

The equation 4x2–3x+7 can be written as 4x2–3x1+7x0

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 4x2–3x+7 is a polynomial in one variable.

**(ii) y2+√2**

Solution:

The equation y2+**√2** can be written as y2+**√**2y0

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 y2+**√**2 is a polynomial in one variable.

**(iii) 3√t+t√2**

Solution:

The equation 3√t+t√2 can be written as 3t1/2+√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) x10+y3+t50**

Solution:

Here, in the equation x10+y3+t50

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

x10+y3+t50. Hence, it is **not **a polynomial in one variable.

**2. Write the coefficients of x2 in each of the following:**

**(i) 2+x2+x**

Solution:

The equation 2+x2+x can be written as 2+(1)x2+x

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

Here, the number that multiplies the variable x2 is 1

, the coefficients of x2 in 2+x2+x is 1.

**(ii) 2–x2+x3**

Solution:

The equation 2–x2+x3 can be written as 2+(–1)x2+x3

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 x2 is -1

the coefficients of x2 in 2–x2+x3 is -1.

**(iii) (/2)x2+x**

Solution:

The equation (/2)x2 +x can be written as (/2)x2 + 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 x2 is /2.

the coefficients of x2 in (/2)x2 +x is /2.

**(iii)√2x-1**

Solution:

The equation √2x-1 can be written as 0x2+√2x-1 [Since 0x2 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 x2is 0

, the coefficients of x2 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., 3x35+5

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

Eg., 4x100

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

**(i) 5x3+4x2+7x**

Solution:

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

Here, 5x3+4x2+7x = 5x3+4x2+7x1

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

the degree of 5x3+4x2+7x is 3 as 3 is the highest power of x in the equation.

**(ii) 4–y2**

Solution:

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

Here, in 4–y2,

The power of the variable y is 2

the degree of 4–y2 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 t 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× x0

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) x2+x**

Solution:

The highest power of x2+x is 2

the degree is 2

Hence, x2+x is a quadratic polynomial

**(ii) x–x3**

Solution:

The highest power of x–x3 is 3

the degree is 3

Hence, x–x3 is a cubic polynomial

**(iii) y+y2+4**

Solution:

The highest power of y+y2+4 is 2

the degree is 2

Hence, y+y2+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) r2**

Solution:

The highest power of r2 is 2

the degree is 2

Hence, r2is a quadratic polynomial.

**(vii) 7x3**

Solution:

The highest power of 7x3 is 3

the degree is 3

Hence, 7x3 is a cubic polynomial.