Problem 16 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$y^{2}+5 y-24$$

Step-by-Step Solution

Answer

The trinomial \(y^2 + 5y - 24\) factors to \((y - 3)(y + 8)\).

Step 1: Identify the Coefficients and the Constant

Here, the coefficients of the terms are 1 (the coefficient of \(y^2\)), 5 (the coefficient of y), and -24 (which is the constant). The goal is to find two numbers that multiply to -24 and add to 5.

Step 2: Find the Numbers

The two numbers that satisfy these conditions are 8 and -3 because \(8 \times -3 = -24\) and \(8 + -3 = 5\).

Step 3: Factor the Trinomial

Now, use these numbers to write the trinomial as the product of two binomials, \(y^2 + 5y - 24 = (y - 3)(y + 8)\).

Step 4: Verify by FOIL Method

Take the factors (y - 3) and (y + 8), and apply the FOIL method (First, Outer, Inner, Last), which should give us back the original trinomial. This is given by \((y - 3)(y + 8) = y^2 + 8y - 3y - 24 = y^2 + 5y - 24\). This confirms that we factored the trinomial correctly.