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 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) 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 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× 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.

Class 9th Maths Polynomial