Problem 9 - 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. $$3 y^{2}+y-4$$

Step-by-Step Solution

Answer

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

Step 1: Finding the pair of numbers

To solve the trinomial, find two numbers that multiply to -12 (4*-3 = -12) and add to 1 (4 + -3 = 1). These numbers are 4 and -3.

Step 2: Factoring the trinomial

The trinomial can be factored as follows: \(3y^2 + y - 4\) can be written as \(3y^2 + 4y - 3y - 4\). By grouping, we can get \((3y^2 + 4y) - (3y + 4)\). Factor out common terms, which gives: \(y(3y + 4) - 1(3y + 4)\). The trinomial is now factored into \((3y + 4)(y - 1)\).

Step 3: Checking the solution using the FOIL method

To ensure the factoring process has been done correctly, apply the FOIL method to the factored form of the trinomial. The terms should multiply back to give the initial equation. (3y + 4)*(y - 1) = \(3y^2 - 3y + 4y - 4\) = \(3y^2 + y - 4\), which is the original trinomial, confirming the factorization is correct.

Key Concepts

Polynomial Factorization

Polynomial factorization involves breaking down a complex expression into simpler factors that, when multiplied together, yield the original polynomial. This process is similar to breaking down a number into its prime factors in arithmetic. For example, the expression \(3y^2 + y - 4\) can appear daunting at first. However, by recognizing it as a trinomial, we can begin to simplify it.

Trinomials are particularly interesting because they consist of three terms, and the factorization often involves finding two binomials whose product returns to the original trinomial. Factorization can help solve equations or simplify expressions, making it a valuable tool in algebra. The process typically involves these steps:

• Identifying the form, such as quadratic trinomials (like \(ax^2 + bx + c\)).
• Finding pairs of numbers (or terms) whose product equals the constant term and sum matches the coefficient of the middle term.
• Rewriting and grouping terms for easier factoring.
• Checking the factorized form with methods like FOIL to ensure accuracy.

By honing these skills, students can tackle a wide range of polynomial expressions with confidence.

FOIL Method

The FOIL method is a straightforward technique used to multiply two binomials, helping verify polynomial factorization. The acronym FOIL stands for First, Outer, Inner, Last, representing the order of multiplication when dealing with two binomials. For instance, when multiplying the binomials \((3y + 4)\) and \((y - 1)\), use FOIL as follows:

- **First:** Multiply the first terms in each binomial, yielding \(3y \cdot y = 3y^2\).- **Outer:** Multiply the outer terms, producing \(3y \cdot (-1) = -3y\).- **Inner:** Multiply the inner terms, giving \(4 \cdot y = 4y\).- **Last:** Multiply the last terms in each binomial, resulting in \(4 \cdot (-1) = -4\).Add these results together: \(3y^2 - 3y + 4y - 4\). Combine like terms to simplify to \(3y^2 + y - 4\), returning us to the original trinomial.

The FOIL method makes it easy to ensure that the factorization is correct, as it should reconstruct the original expression. Understanding and practicing this method is a great way for students to build confidence in algebraic problem-solving.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators like addition and multiplication. Grasping the concept of algebraic expressions is essential for understanding more complex mathematical topics.

When dealing with expressions like \(3y^2 + y - 4\), students can learn how changes in the coefficients and constants affect the behavior of the expression. It's important to familiarize yourself with terms, such as:

• **Coefficients**, which are numbers multiplying the variables (in \(3y^2\), 3 is the coefficient).
• **Variables**, which are letters representing unknown numbers (like \(y\) in \(y - 1\)).
• **Constants**, which are standalone numbers (such as \(-4\)).

In algebra, the ability to interpret and manipulate expressions involves operations and transformations like addition, subtraction, and division (and in more advanced cases, integration and differentiation). Through these skills, students are empowered to solve equations, graph functions, and model real-world scenarios, paving the way for future studies in science, engineering, and beyond. By mastering algebraic expressions, you become ready for solving a wide array of mathematical challenges.