Question

Find the remainder when x 3 + 3x 2 + 3x + 1 is divided by (i) x + 1 (ii) (iii) x (iv) x + π (v) 5 + 2x

Answer 1

Given :   x³ + 3x² + 3x + 1

(i) Let p(x) = x³ + 3x² + 3x +1 and g(x) = x + 1  

By remainder theorem when p(x) divided by g(x) = x + 1 , the remainder is equal to p(-1).

Now, p(x) = x³ + 3x² + 3x +1

∴ p(-1) = (-1)³ + 3(-1)² + 3(-1) + 1

⇒ p(-1) = -1 + 3 - 3 + 1

⇒ p(-1) = 2 - 2 = 0

⇒ p(-1) = 0  

Hence, the  Required remainder is 0.   (By remainder theorem)

(ii) Let p(x) = x³ + 3x² + 3x +1 and g(x) =x - 1/2

By remaindertheorem when p(x) divided by g(x) = x - 1/2 , the remainder is equal to p(½ ).

Now ,  

∴ p(1/2) = (1/2)³ + 3(1/2)² + 3(1/2) + 1

⇒ p(1/2) = 1/8 + ¾  + 3/2 + 1

⇒ p(1/2) = (1 + 6 + 12 + 8)/8

⇒ p(1/2) = 27/8  

Hence, the  Required remainder is 27/8 .   (By remainder theorem)

(iii) Let p(x) = x³ + 3x² + 3x +1 and g(x) = x  

By remainder theorem when p(x) divided by g(x) = x , the remainder is equal to p(0 ).

Now ,  p(x) = x³ + 3x² + 3x +1

∴ p(0) = (0)³ + 3(0)² + 3(0) + 1

⇒ p(0) = 0 + 0 + 0 + 1

⇒ p(0) = 1

Hence, the  Required remainder is 1.   (By remainder theorem)

(iv) Let p(x) = x³ + 3x² + 3x +1 and g(x) = x + π

By remainder theorem when p(x) divided by g(x) = x + π , the remainder is equal to p(- π ).

Now ,  p(x) = x³ + 3x² + 3x +1

∴ p(-π) = (-π)³ + 3(-π)² + 3(-π) + 1

⇒ p(-π) = -π³ + 3π² - 3π + 1

Hence, the  Required remainder is -π³ + 3π² - 3π + 1.   (By remainder theorem)

(v) Let p(x) = x³ + 3x² + 3x +1 and g(x) = 5 + 2x

By remainder theorem when p(x) divided by g(x) = 5 + 2x , the remainder is equal to p(-5/2).

Now ,  p(x) = x³ + 3x² + 3x +1

∴ p(-5/2) = (-5/2)³ + 3(-5/2)² + 3(-5/2) + 1

⇒ p(-5/2) = -125/8 + 75/4 - 15/2 + 1

⇒ p(-5/2) = (-125 + 150 - 60 + 8)/8

⇒ p(-5/2) = -27/8

Hence, the  Required remainder is -27/8.   (By remainder theorem)

Hope this answer will help you….

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Answer 2

Given:  x³ + 3x² + 3x + 1 divided by (i) x + 1 (ii) (iii) x (iv) x + π (v) 5 + 2x.

(i) x + 1

Let p(x) = x³ + 3x² + 3x +1 and g(x) = x + 1  

By remainder theorem, the remainder is equal to p(-1).

As p(x) = x³ + 3x² + 3x +1

∴ p(-1) = (-1)³ + 3(-1)² + 3(-1) + 1

p(-1) = -1 + 3 - 3 + 1

p(-1) = 2 - 2 = 0

p(-1) = 0  

Hence, the remainder is 0.  ______________________

(ii) x - 1/2

Let p(x) = x³ + 3x² + 3x +1 and g(x) = x - 1/2

By remainder theorem, the remainder is equal to p(½ ).

∴ p(1/2) = (1/2)³ + 3(1/2)² + 3(1/2) + 1

p(1/2) = 1/8 + ¾  + 3/2 + 1

p(1/2) = (1 + 6 + 12 + 8)/8

p(1/2) = 27/8  

Hence, the remainder is 27/8. ______________________

(iii) x

Let p(x) = x³ + 3x² + 3x +1 and g(x) = x  

By remainder theorem, the remainder is equal to p(0).

As, p(x) = x³ + 3x² + 3x + 1

∴ p(0) = (0)³ + 3(0)² + 3(0) + 1

p(0) = 0 + 0 + 0 + 1

p(0) = 1

Hence, the  Required remainder is 1.   ______________________

(iv) x + π

Let p(x) = x³ + 3x² + 3x +1 and g(x) = x + π

By remainder theorem, the remainder is equal to p( - π ).

So,  p(x) = x³ + 3x² + 3x +1

∴ p(-π) = (-π)³ + 3(-π)² + 3(-π) + 1

p(-π) = -π³ + 3π² - 3π + 1

Hence, the remainder is -π³ + 3π² - 3π + 1.

_______________________

(v) 5 + 2x

Let p(x) = x³ + 3x² + 3x +1 and g(x) = 5 + 2x

By remainder theorem, the remainder is equal to p(-5/2).

So, p(x) = x³ + 3x² + 3x +1

∴ p(-5/2) = (-5/2)³ + 3(-5/2)² + 3(-5/2) + 1

p(-5/2) = -125/8 + 75/4 - 15/2 + 1

p(-5/2) = (-125 + 150 - 60 + 8)/8

p(-5/2) = -27/8

Hence, the remainder is -27/8.  _________________________