To factorize the polynomial \( 4 x^4 + 3 x^2 y^2 + 9 y^4 \), we proceed by finding two binomials whose product gives us the original polynomial.
Here’s the step-by-step solution:
1. Identify the Polynomial Structure:
The polynomial in question is:
[tex]\[
4 x^4 + 3 x^2 y^2 + 9 y^4
\][/tex]
Notice that this is a quadratic form in \(x^2\) and \(y^2\): \(a(x^2)^2 + b(x^2)(y^2) + c(y^2)^2\), where \(a = 4\), \(b = 3\), and \(c = 9\).
2. Propose a Factored Form:
We look for two binomials of the form \((A x^2 + B x y + C y^2)(D x^2 + E x y + F y^2)\) such that their product gives the original polynomial.
3. Expand the Binomials:
Expanding \((2 x^2 + 3 x y + 3 y^2)\) and \((2 x^2 - 3 x y + 3 y^2)\):
[tex]\[
(2 x^2 - 3 x y + 3 y^2)(2 x^2 + 3 x y + 3 y^2)
\][/tex]
We expand this by distributing each term in the first binomial by each term in the second binomial.
4. Combine Like Terms:
[tex]\[
= (2 x^2)(2 x^2) + (2 x^2)(3 x y) + (2 x^2)(3 y^2) \\
\quad + (-3 x y)(2 x^2) + (-3 x y)(3 x y) + (-3 x y)(3 y^2) \\
\quad + (3 y^2)(2 x^2) + (3 y^2)(3 x y) + (3 y^2)(3 y^2)
\][/tex]
Simplify each multiplication:
[tex]\[
= 4 x^4 + 6 x^3 y + 6 x^2 y^2 \\
\quad - 6 x^3 y - 9 x^2 y^2 - 9 x y^3 \\
\quad + 6 x^2 y^2 + 9 x y^3 + 9 y^4
\][/tex]
5. Cancel and Combine Like Terms:
Combine the like terms:
[tex]\[
= 4 x^4 + (6 x^3 y - 6 x^3 y) + (6 x^2 y^2 - 9 x^2 y^2 + 6 x^2 y^2) + (-9 x y^3 + 9 x y^3) + 9 y^4 \\
= 4 x^4 + 3 x^2 y^2 + 9 y^4
\][/tex]
So, we have successfully factorized the polynomial:
[tex]\[
4 x^4 + 3 x^2 y^2 + 9 y^4 = (2 x^2 - 3 x y + 3 y^2)(2 x^2 + 3 x y + 3 y^2)
\][/tex]
This is the factorized form of the given polynomial.
