Let's work through the expression [tex]\(4x^4 + 3x^2 y^2 + y^4\)[/tex] step-by-step.
1. Identify individual terms:
- [tex]\(4x^4\)[/tex] is a term involving [tex]\(x\)[/tex] raised to the 4th power and multiplied by 4.
- [tex]\(3x^2 y^2\)[/tex] is a term involving [tex]\(x\)[/tex] squared and [tex]\(y\)[/tex] squared, multiplied by 3.
- [tex]\(y^4\)[/tex] is a term involving [tex]\(y\)[/tex] raised to the 4th power.
2. Combine the terms into a single expression:
The total expression sums up these terms. We will need to handle each separately if we were to substitute values or simplify the expression in another context.
3. Expression:
Bringing these terms together, we get:
[tex]\[
4x^4 + 3x^2 y^2 + y^4
\][/tex]
This expression is a polynomial in terms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. The highest degree term is [tex]\(4x^4\)[/tex], so the polynomial is of degree 4 considering the highest power of a single variable. When considering the multivariable context, the term [tex]\(4x^4\)[/tex] and [tex]\(y^4\)[/tex] are both of degree 4, similarly [tex]\(3x^2 y^2\)[/tex] is of degree 4 as well (2 from [tex]\(x^2\)[/tex] and 2 from [tex]\(y^2\)[/tex]).
So, the polynomial in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] given is:
[tex]\[
4x^4 + 3x^2 y^2 + y^4
\][/tex]
Each term adds a different combination of the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex], which are summed to give this polynomial form.
