We are asked to expand the expression $(a-b-c)^2$.

AlgebraAlgebraic ExpansionPolynomialsBinomial Theorem
2025/4/27

1. Problem Description

We are asked to expand the expression (abc)2(a-b-c)^2.

2. Solution Steps

We can rewrite the expression as follows:
(abc)2=(a(b+c))2(a - b - c)^2 = (a - (b+c))^2
We can use the formula for the square of a difference:
(xy)2=x22xy+y2(x - y)^2 = x^2 - 2xy + y^2
In our case, x=ax = a and y=b+cy = b+c.
So,
(a(b+c))2=a22a(b+c)+(b+c)2(a - (b+c))^2 = a^2 - 2a(b+c) + (b+c)^2
Now, we need to expand (b+c)2(b+c)^2:
(b+c)2=b2+2bc+c2(b+c)^2 = b^2 + 2bc + c^2
Substitute this back into the expression:
a22a(b+c)+(b+c)2=a22ab2ac+b2+2bc+c2a^2 - 2a(b+c) + (b+c)^2 = a^2 - 2ab - 2ac + b^2 + 2bc + c^2
Rearranging the terms, we get:
a2+b2+c22ab2ac+2bca^2 + b^2 + c^2 - 2ab - 2ac + 2bc

3. Final Answer

a2+b2+c22ab2ac+2bca^2 + b^2 + c^2 - 2ab - 2ac + 2bc

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