Part – II MATHEMATICS The remainder when the polynomial 1 + x^2 + x^4 + x^6 + \ldots. + x^22 is divided by 1 + x + x^2 + x^3 + \ldots. + x^11 is -

Correct Answer is : 2(1+x2+x4+….+x10)

2(1 + x^2 + x^4 + \ldots. + x^10)

KVPY SX SB Exam 2016 Question Paper

Question : 81 of 120

Marks: +1, -0

Part – II MATHEMATICS

The remainder when the polynomial

1 + x^2 + x^4 + x^6 + …. + x^22

is divided by

1 + x + x^2 + x^3 + …. + x^11

is -

Solution:

P(x) = 1 + x^2 + x^4 + x^6 + …… + x^22

= (1 + x^2) (1 + x^4) (1 + x^4 + x^8) (1 - x^4 + x^8)

Q(x) = 1 + x + x^2 + x^3 + …… + x^11

= (1 + x) (1 + x^2) (1 + x^4 + x^8)

∴{P(x)}/{Q(x)}={1+x^12}/{1+x}

Remainder. When

(1 + x^12)

is divided by (1 + x) is = 2
Now remainder P(x) divided by Q(x)

= 2(1 + x^2) (1 + x^4 + x^8)

= 2(1 + x^2 + ……. + x^10)

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