Certainly! Let's factor the polynomial expression [tex]\( x^4 + x^2 y^2 + y^4 \)[/tex]:
1. Identify Polynomial Structure: Notice that the given polynomial, [tex]\( x^4 + x^2 y^2 + y^4 \)[/tex], is symmetric in [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and can be thought of in terms of [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex].
2. Look for Patterns: By examining the polynomial, we can see it resembles the sum of cubes if we group it carefully. Let's explore possible factorizations to find the specific pattern:
3. Symmetry Observations: The expression [tex]\( x^4 + x^2 y^2 + y^4 \)[/tex] can be viewed using the identity for the sum of cubes broken into symmetrical parts of square terms. Notice how the polynomial can be expressed as products involving polynomials of [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex].
4. Combination of Terms: Let's guess the factorization structure. We recall that certain symmetric polynomials can be factored into symmetrical parts.
Consider:
[tex]\[
(x^2 - x y + y^2)(x^2 + x y + y^2)
\][/tex]
5. Verification: To ensure correctness, let's expand the presumed factorization:
[tex]\[
(x^2 - x y + y^2)(x^2 + x y + y^2)
\][/tex]
We multiply the terms:
[tex]\[
= x^2(x^2 + x y + y^2) - x y(x^2 + x y + y^2) + y^2(x^2 + x y + y^2)
\][/tex]
[tex]\[
= x^4 + x^3 y + x^2 y^2 - x^3 y - x^2 y^2 - x y^3 + x^2 y^2 + x y^3 + y^4
\][/tex]
6. Simplification: Combine the like terms:
[tex]\[
x^4 + (x^3 y - x^3 y) + (x^2 y^2 - x^2 y^2 + x^2 y^2) - x y^3 + x y^3 + y^4 + (x y^3 - x y^3)
\][/tex]
[tex]\[
= x^4 + x^2 y^2 + y^4
\][/tex]
Thus, the given polynomial [tex]\( x^4 + x^2 y^2 + y^4 \)[/tex] can be factored as:
[tex]\[
(x^2 - x y + y^2)(x^2 + x y + y^2)
\][/tex]
Therefore, the resolved factors of the polynomial [tex]\( x^4 + x^2 y^2 + y^4 \)[/tex] are:
[tex]\[
(x^2 - x y + y^2)(x^2 + x y + y^2)
\][/tex]
