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Chapter 0: Problem 103

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

Short Answer

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

Step by step solution

01

Identify a, b, and c

In this problem, a = 3, b = 10, and c = 8 from the quadratic trinomial \(3x^2 + 10x + 8\).
02

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.
03

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: \(3x^2 + 4x + 6x + 8\).
04

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 \(2(3x + 4)\).
05

Rewrite the factored form

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

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