Let's find the product of the expressions [tex]\((x + y)\)[/tex], [tex]\((x - y)\)[/tex], and [tex]\((x^2 + y^2)\)[/tex].
Step-by-Step Solution:
1. Identify the expressions:
- The first expression is [tex]\((x + y)\)[/tex].
- The second expression is [tex]\((x - y)\)[/tex].
- The third expression is [tex]\((x^2 + y^2)\)[/tex].
2. Write the product of these expressions:
[tex]\[
(x + y) \cdot (x - y) \cdot (x^2 + y^2)
\][/tex]
3. Multiply the expressions two at a time:
Let's start with multiplying [tex]\((x + y)\)[/tex] and [tex]\((x - y)\)[/tex]:
[tex]\[
(x + y)(x - y)
\][/tex]
The product of [tex]\((x + y)\)[/tex] and [tex]\((x - y)\)[/tex] is a difference of squares:
[tex]\[
(x + y)(x - y) = x^2 - y^2
\][/tex]
4. Multiply the result by the third expression:
Now multiply [tex]\(x^2 - y^2\)[/tex] by [tex]\((x^2 + y^2)\)[/tex]:
[tex]\[
(x^2 - y^2)(x^2 + y^2)
\][/tex]
Distribute [tex]\((x^2 - y^2)\)[/tex] to both terms in [tex]\((x^2 + y^2)\)[/tex]:
[tex]\[
(x^2 - y^2)(x^2 + y^2) = x^2(x^2 + y^2) - y^2(x^2 + y^2)
\][/tex]
[tex]\[
= (x^2 \cdot x^2 + x^2 \cdot y^2 - y^2 \cdot x^2 - y^2 \cdot y^2)
\][/tex]
[tex]\[
= x^4 + x^2 y^2 - x^2 y^2 - y^4
\][/tex]
Simplify the expression by combining like terms:
[tex]\[
= x^4 - y^4
\][/tex]
Therefore, the product of the expressions [tex]\((x + y)\)[/tex], [tex]\((x - y)\)[/tex], and [tex]\((x^2 + y^2)\)[/tex] is:
[tex]\[
(x - y)(x + y)(x^2 + y^2)
\][/tex]
Final simplified form:
[tex]\[
(x - y)(x + y)(x^2 + y^2)
\][/tex]
