In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.15x² + 111xy − 14y²

Verified step by step guidance

1. Step 1: Identify the trinomial to factor. The given trinomial is $15x^2 + 111xy - 14y^2$.
2. Step 2: Look for a common factor in all terms. In this case, there is no common factor other than 1.
3. Step 3: Use the method of factoring by grouping. First, multiply the leading coefficient (15) by the constant term (-14) to get -210.
4. Step 4: Find two numbers that multiply to -210 and add to the middle coefficient (111). These numbers are 120 and -9.
5. Step 5: Rewrite the middle term using the numbers found: $15x^2 + 120xy - 9xy - 14y^2$. Then, group the terms: $(15x^2 + 120xy) + (-9xy - 14y^2)$. Factor each group separately.

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 often 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.

FOIL Method

The FOIL method is a technique used to multiply two binomials, standing for First, Outside, Inside, Last. This method helps verify the correctness of a factorization by ensuring that the resulting expression matches the original trinomial. Mastery of FOIL is essential for confirming that the factors obtained are accurate.

Prime Trinomials

A prime trinomial is a quadratic expression that cannot be factored into simpler binomials with rational coefficients. Recognizing when a trinomial is prime is important, as it indicates that no integer solutions exist for the factorization. This concept helps in determining the limits of factorization and understanding the nature of quadratic equations.