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Class 9 Maths Chapter 2 HOTS Questions - Polynomials

Q.1. If  p(x) = x2 – 2√2 x + 1, then find  p (2√2) .

Sol. Since, p(x) = x– 2 √2 x + 1  
Class 9 Maths Chapter 2 HOTS Questions - Polynomials
= 4 (2) – 4 (2) + 1 = 8 – 8 + 1 = 1


Q.2. If  a + b + c = 9, and ab + bc + ca = 26, find a+ b2 + c2

Sol. (a + b + c)2 = (a+ b+ c2) + 2 (ab + bc + ca)
⇒ (9)= (a2 + b+ c2) + 2 (26)  = (a+ b2 + c2) + 52
⇒ a2 + b+ c2 =92– 52 = 81– 52 = 29


Q.3. Factorise :Class 9 Maths Chapter 2 HOTS Questions - Polynomials

Sol.   Class 9 Maths Chapter 2 HOTS Questions - Polynomials

Class 9 Maths Chapter 2 HOTS Questions - PolynomialsClass 9 Maths Chapter 2 HOTS Questions - PolynomialsClass 9 Maths Chapter 2 HOTS Questions - PolynomialsClass 9 Maths Chapter 2 HOTS Questions - PolynomialsClass 9 Maths Chapter 2 HOTS Questions - Polynomials


Q.4. If a, b, c are all non-zero and a + b + c = 0, prove that Class 9 Maths Chapter 2 HOTS Questions - Polynomials

Sol. ∵ a + b + c = 0  ⇒  a3 + b3 + c3 – 3abc = 0
or a+ b+ c3 = 3abc ⇒ Class 9 Maths Chapter 2 HOTS Questions - Polynomials


Q.5. If Class 9 Maths Chapter 2 HOTS Questions - Polynomialsthen find the value of Class 9 Maths Chapter 2 HOTS Questions - Polynomials

Sol: 
Class 9 Maths Chapter 2 HOTS Questions - Polynomials


Q.6. Factorise: (a – b)+ (b – c)3 + (c – a)3

Sol: Put a – b = x, b – c = y and c – a = z so that
                       x + y + z = 0 ⇒ x3 + y3 + z3 = 3xyz
                       ⇒ (a – b)3 + (b – c)3 + (c – a)3 = 3(a – b) (b – c) (c – a)





Q.7.  Factorise: 14x6 – 45x3y– 14y6

Sol: Let us put x3 = a and y3 = b so that

                       14x6 – 45x3y– 14y6 = 14a2 – 45ab – 14b2

                       = 14a2 – 49ab + 4ab – 14b2 = (2a – 7b) (7a + 2b)

                       ⇒ 14x6 – 45x3y3 – 14y6 = (2x3 – 7y3) (7x3 + 2y3)


Q.8.  Find the product: (x – 3y) (x + 3y) (x2 + 9y2)

Sol: (x – 3y) (x + 3y) (x2 + 9y2) and 

(x2 – 9y2) (x2 + 9y2) = (x4 – 81y4)


Q.9.   If x2 – 3x + 2 divides x3 – 6x2 + ax + b exactly, then find the value of ‘a’ and ‘b’

Sol: x2 – 3x + 2 = (x – 1) (x – 2) ⇒ x3 – 6x+ ax + b is exactly divisible by

                       (x – 1) and (x – 2) also i.e. f(1) = 0 and f(2) = 0

                       Now, f(1) = 0 ⇒ a + b – 5 = 0

                       and f(2) = 0 ⇒ 2a + b – 16 = 0

                       solving (i) and (ii) a = 11 and b = – 6


Q.10. Prove that (a + b + c)3 – a3 – b3 – c3 = 3(a + b) (b + c) (c + a)

Sol. 
Class 9 Maths Chapter 2 HOTS Questions - Polynomials 
Class 9 Maths Chapter 2 HOTS Questions - Polynomials

The document Class 9 Maths Chapter 2 HOTS Questions - Polynomials is a part of the Class 9 Course Mathematics (Maths) Class 9.
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FAQs on Class 9 Maths Chapter 2 HOTS Questions - Polynomials

1. What is a polynomial?
Ans. A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, which are combined using arithmetic operations such as addition, subtraction, multiplication, and division. The degree of a polynomial is the largest power of the variable in the expression.
2. What are the different types of polynomials?
Ans. There are various types of polynomials such as linear polynomial, quadratic polynomial, cubic polynomial, quartic polynomial, quintic polynomial, and so on. A linear polynomial has a degree of 1, a quadratic polynomial has a degree of 2, a cubic polynomial has a degree of 3, a quartic polynomial has a degree of 4, and a quintic polynomial has a degree of 5.
3. How to factorize a polynomial?
Ans. To factorize a polynomial, we need to find its factors. We can start by looking for a common factor among the terms of the polynomial. Then, we can use different methods such as grouping, difference of squares, and sum and difference of cubes to factorize the polynomial.
4. What is the remainder theorem for polynomials?
Ans. The remainder theorem for polynomials states that if a polynomial f(x) is divided by (x-a), then the remainder is equal to f(a). In other words, if we divide a polynomial f(x) by (x-a), the remainder will be the value of the polynomial when we substitute x=a.
5. What is the importance of polynomials in mathematics?
Ans. Polynomials are an essential part of mathematics and have diverse applications in various fields such as physics, engineering, economics, and computer science. They are used to model and solve real-world problems, and also provide a foundation for more advanced topics such as calculus and algebraic geometry. Understanding polynomials is crucial for developing mathematical reasoning and problem-solving skills.
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