Simplifying the Algebraic Expression: $(a+b+c)^2 - a^2 - b^2 - c^2$

The question asks us to find the value of the given expression: $(a + b + c)^2 - a^2 - b^2 - c^2$. To solve this, we need to expand the term $(a + b + c)^2$ and then subtract the other terms.

Expanding $(a+b+c)^2$

We use the algebraic identity for the square of a trinomial, which is:

$\left(x+y+z\right)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$

Applying this identity to $(a+b+c)^2$, we replace $x$ with $a$, $y$ with $b$, and $z$ with $c$:

$\left(a+b+c\right)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

Substituting and Simplifying the Expression

Now, we substitute this expanded form back into the original expression:

Original Expression: $\left(a + b + c\right)^2 - a^2 - b^2 - c^2$

Substitute expansion: $\left(a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\right) - a^2 - b^2 - c^2$

Next, we simplify by combining like terms. Notice that we have $+a^2, +b^2, +c^2$ from the expansion and $-a^2, -b^2, -c^2$ from the other terms. These terms will cancel each other out:

$a^2 - a^2 = 0$

$b^2 - b^2 = 0$

$c^2 - c^2 = 0$

So, the expression simplifies to:

$\left(a^2 - a^2\right) + \left(b^2 - b^2\right) + \left(c^2 - c^2\right) + 2ab + 2bc + 2ca$

$0 + 0 + 0 + 2ab + 2bc + 2ca$

Which gives us:

$2ab + 2bc + 2ca$

We can factor out the common factor of 2 from this expression:

$2(ab + bc + ca)$

Therefore, the value of the expression $(a + b + c)^2 - a^2 - b^2 - c^2$ is $2(ab + bc + ca)$.

Comparing with Options

Let's compare our simplified result with the given options:

• Option 1: $2abc$
• Option 2: $2(ab + bc + ca)$
• Option 3: $2(a + b + c)$
• Option 4: $2ab + bc - 2ca$

Our simplified expression $2(ab + bc + ca)$ matches Option 2.

Step-by-Step Solution Summary

1. Write down the given expression: $(a + b + c)^2 - a^2 - b^2 - c^2$.
2. Recall the expansion formula for $(a+b+c)^2$: $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$.
3. Substitute the expansion into the expression: $(a^2 + b^2 + c^2 + 2ab + 2bc + 2ca) - a^2 - b^2 - c^2$.
4. Cancel out the $a^2$, $b^2$, and $c^2$ terms.
5. The remaining terms are $2ab + 2bc + 2ca$.
6. Factor out the common factor 2: $2(ab + bc + ca)$.

Revision Table: Key Concepts for Algebraic Simplification

Additional Information: Related Algebraic Identities

Understanding common algebraic identities is crucial for simplifying expressions quickly and accurately. Here are a few related identities:

• Square of a Binomial:
  • $\left(a+b\right)^2 = a^2 + 2ab + b^2$
  • $\left(a-b\right)^2 = a^2 - 2ab + b^2$
• Difference of Squares:
  • $a^2 - b^2 = \left(a+b\right)\left(a-b\right)$
• Sum and Difference of Cubes:
  • $a^3 + b^3 = \left(a+b\right)\left(a^2 - ab + b^2\right)$
  • $a^3 - b^3 = \left(a-b\right)\left(a^2 + ab + b^2\right)$
• Cube of a Binomial:
  • $\left(a+b\right)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
  • $\left(a-b\right)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

Mastering these identities can greatly help in solving problems involving expansion and factorization of algebraic expressions.