Certainly! Let's examine the expression step-by-step:
Given:
[tex]\[ x^4 - 3x^2y^2 + 9y^4 \][/tex]
1. Identify Terms:
The expression involves three distinct terms:
- The first term is [tex]\(x^4\)[/tex].
- The second term is [tex]\(-3x^2y^2\)[/tex].
- The third term is [tex]\(9y^4\)[/tex].
2. Break Down the Expression:
a. [tex]\(x^4\)[/tex]:
- This is a pure term involving [tex]\(x\)[/tex] raised to the fourth power. It indicates the fourth-degree polynomial in [tex]\(x\)[/tex].
b. [tex]\(-3x^2y^2\)[/tex]:
- The middle term mixes both variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. It is a product of [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex] multiplied by [tex]\(-3\)[/tex]. This is a cross-product term.
c. [tex]\(9y^4\)[/tex]:
- This is a pure term involving [tex]\(y\)[/tex] raised to the fourth power. It indicates the fourth-degree polynomial in [tex]\(y\)[/tex], scaled by a factor of 9.
3. Combining the Terms:
Each term is clearly a part of a larger polynomial expression. When combined together:
[tex]\[ x^4 - 3x^2y^2 + 9y^4 \][/tex]
we are looking at a polynomial of degree 4 involving two variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
4. Symmetry and Degrees:
Observe that:
- Each term in the expression is of degree 4.
- The expression [tex]\(x^4\)[/tex] is symmetric in [tex]\(x\)[/tex].
- The term [tex]\(-3x^2y^2\)[/tex] involves a symmetric combination of squared terms.
- The expression [tex]\(9y^4\)[/tex] is symmetric in [tex]\(y\)[/tex].
So, the resultant polynomial [tex]\(x^4 - 3x^2y^2 + 9y^4\)[/tex] is a fourth-degree polynomial with mixed terms for [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
This detailed breakdown illustrates how each component contributes to the overall polynomial structure.
