Problem 19 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$x^{2}-2 x-15$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(x^{2}-2x -15\) is \((x - 5)(x + 3)\).

Step 1: Understand the trinomial pattern

Confirm that the given trinomial is in the standard form \(ax^2+bx+c\), where a, b, and c are constants. In this case, the trinomial \(x^2-2x-15\) is already in the standard form. Note the values of a, b, and c: a=1, b=-2, c=-15.

Step 2: Finding factors

The goal here is to find two numbers that multiply to give ac (which is just c in this case, because a is 1) and add to give b. We're looking for two numbers that multiply to -15 and add to -2. The numbers -5 and 3 fit this criteria because -5 * 3 = -15 and -5 + 3 = -2.

Step 3: Write the factor form

Replace the trinomial with the pair of binomials found in step 2. This means that \(x^2-2x-15\) factors into \((x - 5)(x + 3)\).

Step 4: Check the solution

Generally, it is good practice to check the solution. You can do this by re-expanding the pair of binomials to see if you get back to the given trinomial. \((x - 5)(x + 3)\) expands to \(x^{2}-2x-15\), confirming that the factored form is accurate.