Problem 99 - Introductory Algebra for College Students

Question

In factoring \(3 x^{2}-10 x-8,\) a student lists \((3 x-2)(x+4)\) as a possible factorization. Use FOIL multiplication to determine if this factorization is correct. If it is not correct, describe how the correct factorization can quickly be obtained using these factors.

Step-by-Step Solution

Answer

The original factorization of \(3x^{2} -10x - 8\) as \((3x - 2)(x+4)\) is incorrect. The correct factorization is \((3x+2)(x-4)\).

Step 1: Apply the FOIL method

Expand the provided factors \((3x-2)(x+4)\) using FOIL method as follows:First: Multiply the first terms in each binomial: \(3x * x = 3x^{2}\).Outer: Multiply the outer terms in each pair of parentheses: \(3x * 4 = 12x\).Inner: Multiply the inner terms: \(-2 * x = -2x\).Last: Multiply the last terms: \(-2 * 4 = -8\).Sum it all up: \(3x^{2} + 12x - 2x - 8\). Simplified, we get \(3x^{2} + 10x - 8\).

Step 2: Comparing the expressions

Compare the expression obtained in Step 1 with the original quadratic expression \(3x^{2} - 10x - 8\). If they are identical, then the original factorization was correct. However, the expression we got is \(3x^{2} + 10x - 8\) which mean the original factorization was not correct.

Step 3: Correcting the factorization

The original factorization was incorrect. To correct it, return to the factors \((3x-2)(x+4)\). After reviewing these, correct the factorization by changing the sign of the second term in the expression to adjust for the overlooked negative sign. This gives the correct factorization of \((3x+2)(x-4)\). Follow Step 1 and Step 2 to confirm this.