Problem 103 - College Algebra

Question

Explain how to factor \(3 x^{2}+10 x+8\).

Step-by-Step Solution

Answer

The factored form of \(3x^2 + 10x + 8\) is \((3x + 4)(x + 2)\).

Step 1: Identify a, b, and c

In this problem, a = 3, b = 10, and c = 8 from the quadratic trinomial \(3x^2 + 10x + 8\).

Step 2: Find two numbers that multiply to (a*c) and add to b

Multiply a and c. In this case, 3 * 8 = 24. We need to find two numbers that multiply to 24 and add up to b, which is 10. The numbers 4 and 6 fit these conditions because 4 * 6 = 24 and 4 + 6 = 10.

Step 3: Rewrite the quadratic trinomial

Rewrite the middle term (bx) of the quadratic expression (10x) into two terms, using the two numbers found in Step 2. Therefore, it will be rewritten as: \(3x^2 + 4x + 6x + 8\).

Step 4: Group the terms and factor by grouping

Group the terms to become: \((3x^2 + 4x) + (6x + 8)\). For \(3x^2 + 4x\), you can factor out \(x\), resulting in \(x(3x + 4)\). For \(6x + 8\), you can factor out 2, resulting in \(2(3x + 4)\).

Step 5: Rewrite the factored form

Since both terms now contain the common binomial \((3x + 4)\), you can write it as: \((3x + 4)(x + 2)\).