Problem 54 - 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. $$2 a^{2}+5 a b+2 b^{2}$$

Step-by-Step Solution

Answer

The factors of the trinomial \(2a^{2}+5ab+2b^{2}\) are \((2a+5b)\) and \((a+b)\)

Step 1: Identify the trinomial

The trinomial to be factored is given by the expression \(2a^{2}+5ab+2b^{2}\). It is in the form \(Ax^{2} +Bxy + Cy^{2}\), where \(A = 2, B = 5\) and \(C = 2\).

Step 2: Factorize the trinomial

In order to factorize the given trinomial, one must group the terms in pairs and identify the common factors in each pair. The expression can be rewritten as \((a(2a+5b))+ (b(2a+5b))\). Here, \((2a+5b)\) is a common factor. So, the factorized expression would be \((2a+5b)(a+b)\).

Step 3: Verify the factorization with the FOIL method

You can check the factorization by expanding the brackets using the FOIL method. This results in \((2a+5b)(a+b) = 2a^{2}+2ab+5ab+5b^{2}= 2a^{2}+7ab+5b^{2}\), which is the initial trinomial. Thus, the factorization is correct.

Key Concepts

FOIL Method

The FOIL method stands for First, Outer, Inner, Last and refers to a technique used to multiply two binomials—algebraic expressions containing two terms. It's a cornerstone concept in algebra that helps simplify and solve polynomial equations. To apply the FOIL method, multiply the first terms of each binomial, then the terms on the outside, followed by the terms on the inside, and finally the last terms in each binomial.

For example, let's use FOIL on the binomials \( (2a+5b)\) and \( (a+b)\). Multiplying the first terms gives us \(2a \times a = 2a^2\), the outer terms would be \(2a \times b = 2ab\), the inner terms \(5b \times a = 5ab\), and the last terms \(5b \times b = 5b^2\). Adding these up as per the FOIL sequence, we recover the original trinomial: \(2a^2 + 2ab + 5ab + 5b^2 = 2a^2 + 7ab + 5b^2\), matching the given expression.

Becoming comfortable with the FOIL method ensures you correctly multiply binomials and verify factorizations, as demonstrated in the textbook exercise solution.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. They're used to represent real-world and mathematical relationships. In our textbook exercise, the trinomial \(2a^{2}+5ab+2b^{2}\) is an example of a quadratic algebraic expression, as it includes variables raised to the second power.

Understanding the structure of algebraic expressions is critical when performing operations like factorization. For instance, recognizing that \(2a^{2}+5ab+2b^{2}\) can be grouped into pairs allows the identification of common factors and aids in transforming the expression into a product of simpler expressions or binomials. This comprehension is fundamental to working with polynomials and extends to more complex algebraic manipulation.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials, where the result of multiplying the simpler polynomials together gives the original polynomial. This process is used to solve, simplify, and graph polynomial equations. In our textbook example, we are dealing with factoring a trinomial, which is a specific type of polynomial with three terms.

The given trinomial, \(2a^{2}+5ab+2b^{2}\), can be factored by grouping terms and identifying a common binomial factor. Availing of algebraic identities, like the square of a sum \( (x+y)^2 = x^2 + 2xy + y^2 \), can provide a solid starting point. Effective factorization often depends on recognizing patterns within polynomials and utilizing factoring techniques that adhere to algebraic principles. Once factored, you can apply the previously discussed FOIL method to confirm the resulting expressions indeed combine to form the original polynomial.