Factoring trinomials of the form x² + bx + c is a fundamental skill in algebra that allows you to rewrite quadratic expressions as the product of two binomials. This process essentially reverses the distributive property or FOIL (First, Outside, Inside, Last) method used to multiply binomials. When multiplying two binomials such as (x + p)(x + q), the result is x² + (p + q)x + pq. Recognizing this pattern is key to factoring.

To factor a trinomial x² + bx + c, you need to find two numbers that multiply to c and add to b. These two numbers become the constants in the binomials (x + p)(x + q). For example, factoring x² + 10x + 21 involves finding two numbers that multiply to 21 and add to 10, which are 3 and 7. Thus, the factorization is (x + 3)(x + 7).

When the numbers are less straightforward, a systematic approach helps. Start by identifying the values of b and c. Then list all factor pairs of c, considering both positive and negative pairs depending on the signs of b and c. Use a T-chart or organized list to check which pair sums to b. For instance, to factor x² + 3x − 28, find factor pairs of −28: (1, −28), (−1, 28), (2, −14), (−2, 14), (4, −7), and (−4, 7). The pair (−4, 7) sums to 3, so the factorization is (x − 4)(x + 7).

Similarly, for x² − 11x + 30, factor pairs of 30 are (1, 30), (2, 15), (3, 10), and (5, 6). Since the sum must be −11, use (−5, −6), which multiply to 30 and add to −11. The factorization is (x − 5)(x − 6).

Always verify your factorization by expanding the binomials to ensure they produce the original trinomial. This method applies only when the coefficient of x² is 1. For trinomials with different leading coefficients, other factoring techniques are required.

Mastering this factoring technique enhances your ability to solve quadratic equations, simplify expressions, and understand polynomial functions, forming a foundation for more advanced algebraic concepts.

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