Problem 44 - 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. $$3 x^{2}+4 x y+y^{2}$$

Step-by-Step Solution

Answer

The factorization of the trinomial \(3x^{2}+4xy+y^{2}\) is \((\sqrt{3}x + y)^2\).

Step 1: Identify the structure of the trinomial

The given trinomial is \(3x^{2}+4xy+y^{2}\). It has two variables x and y. This trinomial resembles the square of a binomial A^2 + 2AB + B^2 = (A + B)^2 structure. Here, where A could be the square root of \(3x^{2}\) which is \(\sqrt{3}x\), and B could be the square root of \(y^{2}\), which is \(y\).

Step 2: Factor the trinomial

So, checking if \(4xy\) is equal to \(2.\sqrt{3}x.y\), we see that it does, satisfying the 2AB part of the structure. Hence, trinomial can be factored as: \(3x^{2}+4xy+y^{2} = (\sqrt{3}x + y)^2\)

Step 3: Check the factorization using the FOIL method

The FOIL method stands for First, Outer, Inner, and Last. It's a method for multiplying binomials. Let's check the factorization: \n- First terms = \(\sqrt{3}x * \sqrt{3}x = 3x^{2}\) \n- Outer terms = \(\sqrt{3}x * y = \sqrt{3}xy\). \n- Inner terms = \(\sqrt{3}x * y = \sqrt{3}xy\) \n- Last terms = \(y * y = y^{2}\). \nAdding these up, you get the original expression \(3x^{2}+4xy+y^{2}\). This confirms that the factorization is correct.

Key Concepts

FOIL Method

The FOIL method is an essential technique used for multiplying two binomials. Binomials are expressions with two terms, for example, \(a + b\) and \(c + d\). The name 'FOIL' comes from the first letters of four steps needed to multiply the terms in the binomials: First, Outer, Inner, and Last.

• First: Multiply the first terms of each binomial. In \(a + b\) and \(c + d\), this would be \(a \times c\).
• Outer: Multiply the outer terms, which are \(a\) and \(d\).
• Inner: Multiply the inner terms, \(b\) and \(c\).
• Last: Multiply the last terms of each binomial, \(b \times d\).

By adding these results, you get the complete expansion of the two binomials, \(ac + ad + bc + bd\). The FOIL method is very useful for verification purposes when factoring trinomials into binomials. It ensures that when the factored terms are multiplied, they reconstruct the original trinomial expression.

Square of a Binomial

The concept of the square of a binomial is a cornerstone in algebra. Squaring a binomial means multiplying it by itself. For example, if you have a binomial \(A + B\), squaring it would result in \((A + B)^2\).

• The formula for the square of a binomial is \(A^2 + 2AB + B^2\), which shows the expanded form of the binomial squared.
• From this, we see it consists of the square of the first term (\(A^2\)), twice the product of both terms (\(2AB\)), and the square of the second term (\(B^2\)).

Recognizing this pattern is vital in factoring expressions. For example, in an expression like \(3x^{2}+4xy+y^{2}\), identifying \(A\) and \(B\) can reveal the underlying binomial to be squared. This makes factoring the expression straightforward and quick.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations, such as addition, subtraction, multiplication, and division. They are a fundamental part of algebra and are used to represent mathematical relationships and ideas.

• Expressions like \(3x^2 + 4xy + y^2\) are trinomials, which are a type of algebraic expression with three terms.
• Understanding how to manipulate these expressions, such as factoring, is crucial to solving algebraic equations effectively.

In algebra, expressions need to be simplified or factored to find solutions to equations or to understand more about the expression itself. Factoring is a key tool that can reduce expressions into simpler forms or reveal hidden structures, such as when an expression can be represented as the square of a binomial. Mastery of working with algebraic expressions equips students with the skills needed to tackle complex mathematical problems confidently.