Problem 40 - 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. $$9 y^{2}+5 y-4$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(9y^2 + 5y - 4\) is \((3y - 1)(3y + 4)\)

Step 1: Factoring the trinomial

Try different combination of factors of \(9y^2\) (which are \(3y\) and \(3y\)) and factors of \(-4\) (which are 4 and -1) to get a sum of \(5y\). After some trials, we can find that \((3y - 1)\) and \((3y + 4)\) meet this requirement because \(-1 * 3y\) + \(4 * 3y\) = \(5y\). Therefore, the trinomial can be factored as \((3y - 1)(3y + 4)\).

Step 2: Check factorization by FOIL

Use FOIL operation to verify if factorization is correct. FOIL stands for First, Outer, Inner, Last. Multiply the first terms in each binomials, then the outer terms, the inner terms and the last terms, after that sum up these products. Do \((3y * 3y)\) + \((3y * 4)\) + \((-1 * 3y)\) + \((-1 * 4)\). This becomes \(9y^2 + 12y - 3y - 4\), simplifying to \(9y^2 + 5y - 4\), which matches the original trinomial.

Key Concepts

FOIL Method

The FOIL method is a technique used in algebra for multiplying two binomials together. FOIL stands for First, Outer, Inner, and Last which refers to the order in which you multiply the terms of the binomials. It's a systematic approach that ensures you consider each component of the binomials during multiplication.

Let's break it down:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms of the binomials.
• Last: Multiply the last terms of the binomials.

After these multiplications, you combine like terms (if there are any) to get the simplified algebraic expression. For example, given two binomials \(a + b\) and \(c + d\), use FOIL to expand them as follows: \(ac\) (first), \(ad\) (outer), \(bc\) (inner), \(bd\) (last), leading to the expression \(ac + ad + bc + bd\).

As an educational best practice, using the FOIL method helps visualize the steps and understanding the structure of binomial multiplication.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition and subtraction. Understanding how to work with these expressions is fundamental to algebra. A trinomial is a specific type of algebraic expression that has three terms, commonly written in the form \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants and \(x\) represents the variable.

Factoring trinomials is about finding two binomials that when multiplied together, give back the original trinomial. This process is similar to ‘un-multiplying’ or decomposing the trinomial into simpler, multiplicative components. It requires a blend of critical thinking and pattern recognition as you consider multiple factor combinations for the terms \(a\) and \(c\) to achieve the middle term \(b\).

For successful factoring, it starts with recognizing the structure of the trinomial and applying strategies like trial and error or systematic approaches like the 'AC method' to find the correct binomial factors.

Binomial Multiplication

Binomial multiplication refers to the process of multiplying two binomials together. A binomial is an algebraic expression that contains two terms, such as \(x + y\) or \(a - b\). When two binomials are multiplied, each term in the first binomial must be multiplied by each term in the second binomial—a process which can be effectively accomplished using the FOIL method mentioned earlier.

For instance, multiplying \(x + y\) by \(a - b\) would involve the following steps: \(xa - xb + ya - yb\), ensuring that every term in the first binomial multiplies with each term in the second. Learning and practicing binomial multiplication equips students with a foundational skill for algebra, allowing for more complex problem-solving that involves polynomials.

Furthermore, understanding binomial multiplication is critical for factoring trinomials, as it allows students to verify if their factorization is correct. By reversing the process, converting the product of binomials back into a trinomial, students can check their work and deepen their comprehension of algebraic expressions and multiplication.