In Exercises 103–114, factor completely. x^4−5x^2 y^2+4y^4

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process is essential for simplifying expressions and solving equations. In the case of the given polynomial, recognizing patterns such as the difference of squares or perfect square trinomials can aid in the factoring process.

Quadratic Form

The expression x^4−5x^2y^2+4y^4 can be viewed as a quadratic in terms of x^2. By substituting u = x^2, the polynomial transforms into a standard quadratic form, making it easier to apply factoring techniques. This approach allows for the identification of roots and factors more systematically.

Common Factor Extraction

Before factoring a polynomial, it is often useful to check for a common factor among the terms. In the expression x^4−5x^2y^2+4y^4, identifying and factoring out any common terms can simplify the expression, making it easier to factor the remaining polynomial. This step is crucial for ensuring that all factors are accounted for in the final solution.