Polynomials-Important Questions

1.  If then find the value of

     Hint:     

                       

2. Factorise: (a – b)3 + (b – c)3 + (c – a)3

     Hint:     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)

3. Factorise: 14x6 – 45x3y3 – 14y6

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

                       14x6 – 45x3y3 – 14y6 = 14a2 – 45ab – 14b2

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

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

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

     Hint:     (x – 3y) (x + 3y) (x2 + 9y2) and (x2 – 9y2) (x2 + 9y2) = (x4 – 81y4)

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

     Hint:     x2 – 3x + 2 = (x – 1) (x – 2) ⇒ x3 – 6x2 + 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