To solve the given expression [tex]\( x^3 + 3x^2y + 3xy^2 + y^2 \)[/tex], let's analyze and simplify it.
### Step-by-Step Solution:
1. Identify the Expression:
The given expression is [tex]\( x^3 + 3x^2y + 3xy^2 + y^2 \)[/tex].
2. Look for Patterns:
Notice that the given expression closely resembles the expansion of a binomial expression. Recall the binomial theorem:
[tex]\[
(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
\][/tex]
However, in our expression, the last term is [tex]\( y^2 \)[/tex] instead of [tex]\( y^3 \)[/tex].
3. Rewrite and Compare:
Let's compare the binomial expansion [tex]\( (x + y)^3 \)[/tex] and the given expression term-by-term:
[tex]\[
(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
\][/tex]
Our given expression is:
[tex]\[
x^3 + 3x^2y + 3xy^2 + y^2
\][/tex]
We observe that the first three terms [tex]\( x^3, 3x^2y, \)[/tex] and [tex]\( 3xy^2 \)[/tex] match perfectly with the expansion. The discrepancy arises in the last term.
4. Simplification Insight:
It seems the term [tex]\( y^3 \)[/tex] in the full expansion is replaced by [tex]\( y^2 \)[/tex] in our given expression.
### Interpret the Result:
- The expression [tex]\( x^3 + 3x^2y + 3xy^2 + y^2 \)[/tex] is recognized as part of the polynomial manipulations, similar to binomial expansions but with a specific modification in the last term.
Therefore, the given expression in simplified and analyzed form is:
[tex]\[
x^3 + 3x^2y + 3xy^2 + y^2
\][/tex]
This completes the analysis and identification of the given mathematical expression.
