Which one of the following is a factor of a² - b² - c² + 2bc + a + b - c?

Correct Answer is : a - b + c + 1

Correct Answer is : a - b + c + 1

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UPSC CDS 2 2023 Math Paper

Question : 75 of 100

Marks: +1, -0

a^{2} - b^{2} - c^{2} + 2bc + a + b - c

Solution:

Concept:

Factorisation using algebraic identities like

a^2 - (b - c)^2

and grouping.

Explanation:

Start with the expression:

a^2 - b^2 - c^2 + 2bc + a + b - c

Group the terms:

a^2 - (b^2 + c^2 - 2bc) + a + b - c

Recognise

b^2 + c^2 - 2bc = (b - c)^2

So we have:

a^2 - (b - c)^2 + a + b - c

Now factor

a^2 - (b - c)^2

(a + (b - c))(a - (b - c)) = (a + b - c)(a - b + c)

The expression becomes:

(a + b - c)(a - b + c) + (a + b - c)

Factor out the common term

(a + b - c)

(a + b - c)[(a - b + c) + 1] = (a + b - c)(a - b + c + 1)

Thus one factor is

a - b + c + 1

Answer:

a - b + c + 1

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