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Exercise - 2.3

Question-1 :-  Find the remainder when x³ + 3x² + 3x + 1 is divided by
(i) x + 1,    (ii) x - 1/2,    (iii) x,    (iv) x + π,    (v) 5 + 2x.

Solution :-
(i) p(x) = x³ + 3x² + 3x + 1, g(x) = x + 1
     polynomial
   Now, q(x) = x² + 2x + 1, r(x) = 0
  
   Verification : 
   p(x) = g(x)q(x) + r(x)
        = (x + 1)(x² + 2x + 1) + 0
        = x³ + 2x² + x + x² + 2x + 1
        = x³ + 3x² + 3x + 1
(ii) p(x) = x³ + 3x² + 3x + 1, g(x) = x - 1/2 
      polynomial
   Now, q(x) = x² + 7x/2 + 19/4, r(x) = 27/8

   Verification : 
   p(x) = g(x)q(x) + r(x)
        = (x - 1/2)(x² + 7x/2 + 19/4) + 27/8
        = x³ + 7x²/2 + 19x/4 - x²/2 - 7x/4 - 19/8 + 27/8
        = x³ + (7x² - x²)/2 + (19x - 7x)/4 + (-19 + 27)/8
        = x³ + 6x²/2 + 12x/4 + 8/8
        = x³ + 3x² + 3x + 1
     
   
(iii) p(x) = x³ + 3x² + 3x + 1,  g(x) = x
     polynomial
   Now, q(x) = x² + 3x + 3, r(x) = 1
  
   Verification : 
   p(x) = g(x)q(x) + r(x)
        = x(x² + 3x + 3) + 1
        = x³ + 3x² + 3x + 1
        
(iv) p(x) = x³ + 3x² + 3x + 1, g(x) = x + π
     polynomial
   Now, q(x) = x² + (3 - π)x + (3 - 3π + π²), r(x) = 1 - 3π + 3π - π³

   Verification : 
   p(x) = g(x)q(x) + r(x)
        = (x + π)[x² + (3 - π)x + (3 - 3π + π²)] + 1 - 3π + 3π - π³
        = (x + π)[x² + 3x - πx + 3 - 3π + π²] + 1 - 3π + 3π - π³
        = x³ + 3x² - πx² + 3x - 3πx + π²x + πx² + 3πx - xπ² + 3π - 3π² + π³ + 1 - 3π + 3π - π³
        = x³ + 3x² + (-πx² + πx²) + 3x + (-3πx + 3πx) + (π²x - π²x) + (3π - 3π) + (-3π² + 3π²) + (π³ - π³) + 1
        = x³ + 3x² + 3x + 1
     
   
(v) p(x) = x³ + 3x² + 3x + 1, g(x) = 5 + 2x    
   polynomial
   Now, q(x) = x²/2 + x/4 + 7/8, r(x) = -27/8

   Verification : 
   p(x) = g(x)q(x) + r(x)
        = (2x + 5)(x²/2 + x/4 + 7/8) + (-27/8)
        = 2x³/2 + 2x²/4 + 14x/8 + 5x²/2 + 5x/4 + 35/8 - 27/8
        = x³ + x²/2 + 5x²/2 + 7x/4 + 5x/4 + (35 - 27)/8
        = x³ + (x² + 5x²)/2 + (7x + 5x)/4 + 8/8
        = x³ + 6x²/2 + 12x/4 + 1
        = x³ + 3x² + 3x + 1
    

Question-2 :-  Find the remainder when x³ – ax² + 6x – a is divided by x – a.

Solution :-
   p(x) = x³ – ax² + 6x – a, g(x) = x - a
   polynomial
   q(x) = x² + 6, r(x) = 5a
  
   Verification : 
   p(x) = g(x)q(x) + r(x)
        = (x - a)(x² + 6) + 5a
        = x³ + 6x - ax² - 6a + 5a
        = x³ – ax² + 6x – a
    

Question-3 :-  Check whether 7 + 3x is a factor of 3x³ + 7x.

Solution :-
  p(x) = of 3x, g(x) = 7 + 3x
  polynomial
  q(x) = x² - 7x/3 + 70/9, r(x) = -490/9
  Therefore, remainder is not zero. So, 7 + 3x is not a factor of 3x³ + 7x.
    
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