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NCERT Solutions for Class 9 Maths Chapter 2 - Polynomials (Exercise 2.1)

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


(ii) y2+√2
Ans: The equation y+ √2 can be written as y+ √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 y+ √2 is a polynomial in one variable.


(iii) 3√t + t√2
Ans: 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
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 + y+ t50
Ans: Here, in the equation x10 + y+ t50
Though the powers, 10, 3, 50, are whole numbers, there are 3 variables used in the expression
x10 + y+ t50.
Hence, it is not a polynomial in one variable.


Q2. Write the coefficients of x2 in each of the following:
(i) 2 + x+ x
Ans: The equation 2 + x2+x can be written as 2 + (1)x+ 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 + x+ x is 1.


(ii) 2 – x2  + x3
Ans: The equation 2 – x+ x3 can be written as 2 + (–1)x+ 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 – x+ x3 is -1.


(iii) (π/2)x+ 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) 5x+  4x2  + 7x
Ans: The highest power of the variable in a polynomial is the degree of the polynomial.
Here, 5x+ 4x+ 7x = 5x+ 4x+ 7x1
The powers of the variable x are: 3, 2, 1
The degree of 5x+ 4x+ 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
AnsThe 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:
AnsWe 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+ x
AnsThe highest power of x+ x is 2
The degree is 2
Hence, x+ x is a quadratic polynomial


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


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


(iv) 1 + x
AnsThe 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
AnsThe highest power of r2 is 2
The degree is 2
Hence, ris a quadratic polynomial.


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

The document NCERT Solutions for Class 9 Maths Chapter 2 - Polynomials (Exercise 2.1) is a part of the Class 9 Course Mathematics (Maths) Class 9.
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FAQs on NCERT Solutions for Class 9 Maths Chapter 2 - Polynomials (Exercise 2.1)

1. What are polynomials?
Ans. Polynomials are mathematical expressions consisting of variables and coefficients, which involve only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
2. What is the degree of a polynomial?
Ans. The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, the degree of the polynomial 5x^3 + 2x^2 - 3x + 1 is 3.
3. What is the remainder theorem of polynomials?
Ans. The remainder theorem of polynomials states that if a polynomial f(x) is divided by x-a, then the remainder is equal to f(a). This theorem is useful in finding the remainder of a polynomial when it is divided by another polynomial of lower degree.
4. How do we find the roots of a polynomial equation?
Ans. To find the roots of a polynomial equation, we set the polynomial equal to zero and solve for the variable. The roots are the values of the variable that make the polynomial equal to zero. For example, to find the roots of the polynomial x^2 - 4x + 3, we set it equal to zero and solve for x to get x=1 and x=3.
5. What is the factor theorem of polynomials?
Ans. The factor theorem of polynomials states that if a polynomial f(x) has a factor (x-a), then f(a) = 0. This theorem is useful in finding factors of a polynomial and can be used to factorize a polynomial completely.
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