Problem 2 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$3 x^{2}+5 x+2$$

Step-by-Step Solution

Answer

The factorization of the trinomial \(3x^2+5x+2\) is \((3x+2)(x+1)\).

Step 1: Identifying the coefficients

The trinomial given is \(3x^2+5x+2\). Here, the coefficient of \(x^2\) (a) is 3, the coefficient of \(x\) (b) is 5 and the constant term (c) is 2.

Step 2: Finding two numbers

Find two numbers that multiply to \(ac=3*2=6\) and add up to \(b=5\). The numbers are 2 and 3.

Step 3: Rewrite the middle term

Rewrite the trinomial, expressing the middle term (5x) as the sum of the terms 2x+3x, which gives \(3x^2+2x+3x+2\).

Step 4: Factor by grouping

Group the terms to get \(x(3x+2)+1(3x+2)\). The factorized form becomes \((3x+2)(x+1)\).

Step 5: Check with FOIL multiplication

To validate the factoring, multiply the factors using the FOIL method. The First terms are \(3x*x=3x^2\), Outer terms are \(3x*1=3x\), Inner terms are \(2*x=2x\), Last terms are \(2*1=2\). Adding these results gives the original trinomial \(3x^2+5x+2\).

Key Concepts

Understanding Trinomials

A trinomial is a type of polynomial with exactly three terms. It usually takes the form:

• \( ax^2 + bx + c \,\)

where \(a, b,\) and \(c\) are constants and \(x\) is a variable. Trinomials are common in algebra and are the starting point for factoring exercises.
The primary task with a trinomial is to break it down or "factor" it into simpler polynomial expressions, which, when multiplied together, give back the original trinomial.
Understanding the relation between multiplication and factors is key in algebra, as it helps simplify expressions and solve equations.
There are a range of techniques for factoring trinomials, such as the "factoring by grouping" technique, which we'll discuss later.
By recognizing the specific form of a trinomial, you can decide the best method to tackle the problem efficiently.

Applying the FOIL Method

The FOIL method is an acronym used to simplify the process of multiplying two binomials.

• **F** - Firsts: Multiply the first terms of each binomial.
• **O** - Outers: Multiply the outer terms.
• **I** - Inners: Multiply the inner terms.
• **L** - Lasts: Multiply the last terms.

This method is particularly useful for verifying the results after factoring a trinomial, ensuring that the multiplication of your factors results in the original expression.
For example, when checking \( (3x+2)(x+1)\) for the trinomial \(3x^2 + 5x + 2\), each binomial part involved in FOIL confirms that the factorization is correct once summed back.
The FOIL method offers a straightforward way to expand binomials, especially when ensuring your factored expression is valid.

Mastering the Grouping Method

The grouping method is a powerful technique for factoring trinomials, especially useful when dealing with expressions where a simple trial and error approach might fail or slow down the process.
To factor a trinomial using the grouping method, follow these steps:

• Identify the product of the first and last coefficients (i.e., \(a \times c\)).
• Find two numbers that multiply to this product and add up to the middle coefficient \(b\).
• Rewrite the middle term using these two numbers.
• Group the terms into pairs and factor each pair.
• Factor out the common binomial.

This method effectively transforms a trinomial into an easily factorable form by introducing an intermediate step, making it manageable.
For instance, in the trinomial \(3x^2+5x+2\), you find that 2 and 3 multiply to 6 and add up to 5.
Rewriting 5x as 2x + 3x allows you to factor by grouping and systematically break it down.
With practice, the grouping method becomes a versatile tool in your algebra toolkit.