In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.x² + xy + y²

Verified step by step guidance

1. Identify the trinomial: \(x^2 + xy + y^2\).
2. Check if the trinomial can be factored by looking for two binomials \((ax + by)(cx + dy)\) that multiply to give the original trinomial.
3. Consider the structure of the trinomial: it resembles a perfect square trinomial, which is typically of the form \((x + y)^2 = x^2 + 2xy + y^2\).
4. Notice that the middle term \(xy\) is not twice the product of \(x\) and \(y\), which suggests that this trinomial might not be a perfect square.
5. Conclude that \(x^2 + xy + y^2\) is a prime trinomial because it cannot be factored into simpler polynomials with integer coefficients.

Key Concepts

Factoring Trinomials

Factoring trinomials involves rewriting a quadratic expression in the form ax² + bx + c as a product of two binomials. This process requires identifying two numbers that multiply to ac (the product of a and c) and add to b. Understanding this concept is crucial for simplifying expressions and solving equations.

Prime Trinomials

A prime trinomial is a quadratic expression that cannot be factored into the product of two binomials with rational coefficients. Recognizing when a trinomial is prime is essential, as it indicates that the expression cannot be simplified further. This concept helps in determining the nature of the roots of the quadratic equation.

FOIL Method

The FOIL method is a technique used to multiply two binomials, standing for First, Outside, Inside, Last, which refers to the order in which the terms are multiplied. This method is also useful for checking the accuracy of a factorization by ensuring that the product of the binomials returns to the original trinomial. Mastery of FOIL is important for verifying solutions in algebra.