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UPSC CDS I 2025 Mathematics Paper

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Question : 83 of 100

Marks: +1, -0

what is the Hcf of the polynomials

x^{8} + x^{4} + 1

x^{4} + x^{2} + 1

?

1
$x^{4} - x^{2} + 1$
x⁴ + x² + 1
$x^{4} - x^{2} - 1$

Solution:

Concept:

Use the identity

a^2 + ab + b^2 = (a+b)^2 - ab

to factor the given polynomials.

Explanation:

Write

f(x) = x^8 + x^4 + 1 = (x^4)^2 + x^4 \cdot 1 + 1^2

which fits

a^2 + ab + b^2

with

a = x^4

,

b = 1

.

Similarly,

g(x) = x^4 + x^2 + 1 = (x^2)^2 + x^2 \cdot 1 + 1^2

with

a = x^2

,

b = 1

.

Apply the identity:

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

.

Thus

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

and

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

.

The common factor of both is

x^4 + x^2 + 1

.

Answer:

x^4 + x^2 + 1

(Option C)

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