Problem 13 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}+3 x-10$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(x^{2} + 3x - 10\) is \((x - 2)(x + 5)\).

Step 1: Identify the coefficients

From the given trinomial \(x^{2}+3x-10\), the coefficient of \(x^{2}\) (a) is 1, the coefficient of \(x\) (b) is 3 and the constant term (c) is -10.

Step 2: Find the factors

Look for two numbers that multiply to give -10 (the product of a and c) and add to give 3 (b). In this case, the two numbers are -2 and 5 since \(-2 * 5 = -10\) and \(-2 + 5 = 3\).

Step 3: Write the factored form

Write the trinomial in its factored form using the numbers found. Hence the factored form is \((x - 2)(x + 5)\).

Step 4: Verification Using FOIL

To verify, multiply out \((x - 2)(x + 5)\) using the FOIL method: First terms: \(x * x = x^{2}\), Outside terms: \(x * 5 = 5x\), Inside terms: \(-2 * x = -2x\), Last terms: \(-2 * 5 = -10\). Summing all terms, we get \(x^{2} + 3x - 10\), which is exactly the original trinomial.