Problem 1 - 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. $$2 x^{2}+5 x+3$$

Step-by-Step Solution

Answer

The factored expression for the trinomial \(2x^{2}+5x+3\) is \((x+1)(2x+3)\).

Step 1: Setting Up the Equation

Consider the trinomial \(2x^{2}+5x+3\). The given trinomial which is in the form of \(ax^{2}+bx+c\) where \(a = 2\), \(b = 5\) and \(c = 3\).

Step 2: Factoring the Trinomial

Now factor the trinomial. This can be done by finding two numbers that multiply to \(a \times c = 2 \times 3 = 6\) and add up to \(b\). The numbers that satisfy this condition are \(3\) and \(2\). So, the trinomial can be factored as: \(2x^{2}+3x+2x+3\) which simplifies to \((2x^{2}+3x)+(2x+3)\). By taking common factors out, the expression can be further simplified to \(x(2x+3)+1(2x+3)\).

Step 3: Final Factorization

In this step, the two expressions with common factor \((2x+3)\) are combined to give the final factorization. Hence the factored form of given trinomial will be \((x+1)(2x+3)\).

Step 4: Verification using FOIL multiplication

To validate the factored form, use FOIL multiplication technique. FOIL stands for First, Outer, Inner and Last. When multiplying \((x+1)(2x+3)\), the 'First' terms x and 2x give \(2x^{2}\), 'Outer' terms x and 3 give \(3x\), 'Inner' terms 1 and 2x give \(2x\) and 'Last' terms 1 and 3 give 3. Adding these all turns out to the original trinomial \(2x^{2}+5x+3\).

Key Concepts

FOIL Multiplication

FOIL multiplication is a useful method for multiplying two binomials to get a polynomial. The acronym FOIL stands for First, Outer, Inner, Last, which represent the order in which you multiply the terms in each binomial.

Understanding FOIL

When we have a product like \( (x+a)(x+b) \), we first multiply the First terms of each binomial: \( x \times x \). Then move to the Outer terms: \( x \times b \), and Inner: \( a \times x \). Finally, we multiply the Last terms: \( a \times b \). These products are then added together to get the expanded polynomial expression.

Applying FOIL in Exercises

In the textbook exercise, after finding the factors of the trinomial \( 2x^2 + 5x + 3 \), we can check the factorization using FOIL. When we FOIL the factors \( (x+1)(2x+3) \), it simplifies back to the original trinomial, confirming that our factorization is correct.

Polynomial Expressions

Polynomial expressions are algebraic equations that contain multiple terms which can include constants, variables, and exponents. These expressions can have various degrees based on the highest exponential value of the variable.

Components of Polynomials

With trinomials like \( 2x^2 + 5x + 3 \), we're dealing with polynomial expressions that consist of three terms. The degree of the polynomial is determined by the term with the highest exponent, which in this case is 2, making it a second-degree polynomial.

Importance in Algebra

Understanding the structure and degrees of polynomial expressions is vital as it helps in performing various operations such as addition, subtraction, and particularly factorization, which is often essential in solving algebraic equations.

Algebraic Factorization

Algebraic factorization involves breaking down a complicated algebraic expression into simpler components or 'factors' that, when multiplied together, give back the original expression.

Factorizing Trinomials

For the trinomial \( 2x^2 + 5x + 3 \), the factorization process involves finding two binomials whose product returns the original trinomial. This step can be challenging, but by applying techniques like finding two numbers that multiply to the product of the coefficient of \( x^2 \) and the constant term, and that also add up to the coefficient of the linear term, we can determine the necessary factors.

Application to the Exercise

In our exercise, we end up with the factors \( (x+1)(2x+3) \). The process demonstrates not just the power of algebraic factorization for simplifying expressions but also for solving algebraic equations and inequalities, which is a foundational skill in algebra.