Q.1. If p(x) = x^{2} – 2√2 x + 1, then find p (2√2) .
Sol. Since, p(x) = x^{2 }– 2 √2 x + 1
= 4 (2) – 4 (2) + 1 = 8 – 8 + 1 = 1
Q.2. If a + b + c = 9, and ab + bc + ca = 26, find a^{2 }+ b^{2} + c^{2}
Sol. (a + b + c)^{2} = (a^{2 }+ b^{2 }+ c^{2}) + 2 (ab + bc + ca)
⇒ (9)^{2 }= (a^{2} + b^{2 }+ c^{2}) + 2 (26) = (a^{2 }+ b^{2} + c^{2}) + 52
⇒ a^{2} + b^{2 }+ c^{2} =9^{2}– 52 = 81– 52 = 29
Q.3. Factorise :
Sol.
Q.4. If a, b, c are all nonzero and a + b + c = 0, prove that
Sol. ∵ a + b + c = 0 ⇒ a^{3} + b^{3} + c^{3} – 3abc = 0
or a^{3 }+ b^{3 }+ c^{3} = 3abc ⇒
Q.5. If then find the value of
Sol:
Q.6. Factorise: (a – b)^{3 }+ (b – c)^{3} + (c – a)^{3}
Sol: Put a – b = x, b – c = y and c – a = z so thatx + 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: 14x^{6} – 45x^{3}y^{3 }– 14y^{6}
Sol: Let us put x^{3} = a and y^{3} = b so that
14x^{6} – 45x^{3}y^{3 }– 14y^{6} = 14a^{2} – 45ab – 14b^{2}
= 14a^{2} – 49ab + 4ab – 14b^{2} = (2a – 7b) (7a + 2b)
⇒ 14x^{6} – 45x^{3}y^{3} – 14y^{6} = (2x^{3} – 7y^{3}) (7x^{3} + 2y^{3})
Q.8. Find the product: (x – 3y) (x + 3y) (x^{2} + 9y^{2})
Sol: (x – 3y) (x + 3y) (x^{2} + 9y^{2}) and
(x^{2} – 9y^{2}) (x^{2} + 9y^{2}) = (x^{4} – 81y^{4})
Q.9. If x^{2} – 3x + 2 divides x^{3} – 6x^{2} + ax + b exactly, then find the value of ‘a’ and ‘b’
Sol: x^{2} – 3x + 2 = (x – 1) (x – 2) ⇒ x^{3} – 6x^{2 }+ 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} – a^{3} – b^{3} – c^{3} = 3(a + b) (b + c) (c + a)
Sol.
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Short Notes: Polynomials Doc  3 pages 
Quick Revision: Polynomials Video  09:53 min 
Important Formulas: Polynomials Doc  2 pages 
1. What is a polynomial? 
2. What are the different types of polynomials? 
3. How to factorize a polynomial? 
4. What is the remainder theorem for polynomials? 
5. What is the importance of polynomials in mathematics? 
Short Notes: Polynomials Doc  3 pages 
Quick Revision: Polynomials Video  09:53 min 
Important Formulas: Polynomials Doc  2 pages 

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