(a+b+c+d)(a^2+b^2+c^2+d^2-ab-ac-ad-bc-bd-cd)

Jasonbradley

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Jun 16, 2024

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Expanding the Expression (a+b+c+d)(a^2+b^2+c^2+d^2-ab-ac-ad-bc-bd-cd)

This expression is a product of two factors:

• (a+b+c+d): This is a simple sum of four variables.
• (a^2+b^2+c^2+d^2-ab-ac-ad-bc-bd-cd): This is a more complex expression containing squares and products of the variables.

To expand this expression, we need to distribute each term in the first factor over the second factor. This can be done using the distributive property of multiplication.

Expanding the Expression

We will multiply each term in (a+b+c+d) with each term in (a^2+b^2+c^2+d^2-ab-ac-ad-bc-bd-cd):

1. Multiplying 'a' with the second factor:

a(a^2+b^2+c^2+d^2-ab-ac-ad-bc-bd-cd) = a^3 + ab^2 + ac^2 + ad^2 - a^2b - a^2c - a^2d - abc - abd - acd

2. Multiplying 'b' with the second factor:

b(a^2+b^2+c^2+d^2-ab-ac-ad-bc-bd-cd) = ba^2 + b^3 + bc^2 + bd^2 - ab^2 - abc - abd - b^2c - b^2d - bcd

3. Multiplying 'c' with the second factor:

c(a^2+b^2+c^2+d^2-ab-ac-ad-bc-bd-cd) = ca^2 + cb^2 + c^3 + cd^2 - abc - ac^2 - acd - bc^2 - bcd - c^2d

4. Multiplying 'd' with the second factor:

d(a^2+b^2+c^2+d^2-ab-ac-ad-bc-bd-cd) = da^2 + db^2 + dc^2 + d^3 - abd - acd - ad^2 - bcd - bd^2 - cd^2

Combining all the terms:

Now, we add all the terms obtained from multiplying each term of the first factor with the second factor:

a^3 + ab^2 + ac^2 + ad^2 - a^2b - a^2c - a^2d - abc - abd - acd + ba^2 + b^3 + bc^2 + bd^2 - ab^2 - abc - abd - b^2c - b^2d - bcd + ca^2 + cb^2 + c^3 + cd^2 - abc - ac^2 - acd - bc^2 - bcd - c^2d + da^2 + db^2 + dc^2 + d^3 - abd - acd - ad^2 - bcd - bd^2 - cd^2

Simplifying the Expression:

After combining like terms, the simplified form of the expanded expression is:

a^3 + b^3 + c^3 + d^3 - 3(a^2b + a^2c + a^2d + ab^2 + ac^2 + ad^2 + b^2c + b^2d + bc^2 + bd^2 + c^2d + cd^2) + 6abcd

Key Observations:

• This expanded expression demonstrates a pattern related to the sum of cubes and the sum of squares.
• The expression highlights the importance of careful distribution and combining like terms when expanding complex algebraic expressions.

Understanding the expansion of this expression can be helpful in various mathematical contexts, including algebraic manipulations, factorization, and simplifying equations.