Question

1. Find the remainder when x square 3 3x square 2 + 3x + 1 is divided by
(1) x+1​

Answer 1

Probable Question1 :

Find remainder obtained when the polynomial x³ + 3x² + 3x + 1 is divided by (x + 1) .

Answer :

Remainder , R = 0

Solution :

Here ,

The given polynomial is ; x³ + 3x² + 3x + 1

Now ,

Let the given polynomial be p (x) .

Thus ,

p(x) = x³ + 3x² + 3x + 1

Now ,

According to the remainder theorem ,

If the given polynomial p(x) is divided by (x + 1) , then the remainder R will be p(-1) .

[ If x + 1 = 0 , then x = -1 ]

Thus ,

=> R = p(-1)

=> R = (-1)³ + 3•(-1)² + 3•(-1) + 1

=> R = - 1 + 3 - 3 + 1

=> R = 0

Hence ,

Remainder , R = 0 .

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Probable Question2 :

Find remainder obtained when the polynomial x³ - 3x² + 3x + 1 is divided by (x + 1) .

Answer :

Remainder , R = -6

Solution :

Here ,

The given polynomial is ; x³ - 3x² + 3x + 1

Now ,

Let the given polynomial be p (x) .

Thus ,

p(x) = x³ - 3x² + 3x + 1

Now ,

According to the remainder theorem ,

If the given polynomial p(x) is divided by (x + 1) , then the remainder R will be p(-1) .

[ If x + 1 = 0 , then x = -1 ]

Thus ,

=> R = p(-1)

=> R = (-1)³ - 3•(-1)² + 3•(-1) + 1

=> R = - 1 - 3 - 3 + 1

=> R = -6

Hence ,

Remainder , R = -6 .

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Note :

★ Remainder theorem : If a polynomial p(x) is divided by (x - c) , then the remainder obtained is given as R = p(c) .

★ Factor theorem :

If the remainder obtained on dividing a polynomial p(x) by (x - c) is zero , ie. if R = p(c) = 0 , then (x - c) is a factor of the polynomial p(x) .

If (x - c) is a factor of the polynomial p(x) , then the remainder obtained on dividing the polynomial p(x) by (x - c) is zero , ie. R = p(c) = 0 .