Chapter 2.1 POLYNOMIALS Solutions

Q1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) 4x²–3x+7
Ans: 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
Ans: 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
Ans: 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
Ans: The equation y + 2/y can 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
Ans: 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.

Q2. Write the coefficients of x² in each of the following:
(i) 2 + x² + x
Ans: 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
Ans: 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) (π/2)x² + x
Ans: The equation (π/2)x² + x can be written as (π/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 π/2.
the coefficients of x² in (π/2)x² +x is π/2.

(iv)√2x – 1
Ans: 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 x² is 0, the coefficients of x² in √2x – 1 is 0.

Q3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Ans: The degree of a polynomial is the highest power of the variable in the polynomial. It represents the highest exponent of the variable within the algebraic expression.
Therefore,

• Binomial of degree 35: A polynomial having two terms and the highest degree 35 is called a binomial of degree 35
  Example: 3x³⁵+5
• Monomial of degree 100: A polynomial having one term and the highest degree 100 is called a monomial of degree 100
  Example: 4x¹⁰⁰

Q4. Write the degree of each of the following polynomials:
(i) 5x³ +  4x²  + 7x
Ans: 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
Ans: 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
Ans: 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
Ans: 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.

Q5. Classify the following as linear, quadratic and cubic polynomials:
Ans: 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
Ans: The highest power of x² + x is 2
The degree is 2
Hence, x² + x is a quadratic polynomial

(ii) x – x³
Ans: The highest power of x–x³ is 3
The degree is 3
Hence, x–x³ is a cubic polynomial

(iii) y + y² + 4
Ans: The highest power of y+y²+4 is 2
The degree is 2
Hence, y+y²+4is a quadratic polynomial

(iv) 1 + x
Ans: The highest power of 1 + x is 1
The degree is 1
Hence, 1 + x is a linear polynomial.

(v) 3t
Ans: The highest power of 3t is 1
The degree is 1
Hence, 3t is a linear polynomial.

(vi) r²
Ans: The highest power of r² is 2
The degree is 2
Hence, r² is a quadratic polynomial.

(vii) 7x³
Ans: The highest power of 7x³ is 3
The degree is 3
Hence, 7x³ is a cubic polynomial.