Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ (3 x+2 y)^{2} $$

Step-by-Step Solution

Answer

The polynomial is \(9x^2 + 12xy + 4y^2\).

Step 1: Identify the expression to be expanded

The given expression is \((3x + 2y)^2\). We need to expand this expression to express it as a polynomial.

Step 2: Recognize the format of the expression

The expression \((3x + 2y)^2\) is in the form of \((a + b)^2\). The formula for expanding this form is \(a^2 + 2ab + b^2\).

Step 3: Apply the formula to expand the expression

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), identify \(a = 3x\) and \(b = 2y\). Substitute these into the formula: \((3x)^2 + 2(3x)(2y) + (2y)^2\).

Step 4: Calculate each term in the expansion

Calculate each component of the expanded expression: - \((3x)^2 = 9x^2\) - \(2(3x)(2y) = 12xy\) - \((2y)^2 = 4y^2\).

Step 5: Write down the expanded polynomial

Combine the calculated terms to write the expression as a polynomial: \(9x^2 + 12xy + 4y^2\).

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra that provides a formula for expanding expressions that are raised to a power, like our example \((3x + 2y)^2\). This theorem is particularly helpful when dealing with binomials, which are expressions consisting of two terms. According to the binomial theorem, all expressions of the form \((a + b)^n\) can be expanded into a sum involving terms of a certain power of both \(a\) and \(b\).

The general formula given by the binomial theorem for any positive integer \(n\) is:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

Here, \(\binom{n}{k}\) is the binomial coefficient, which calculates combinations of \(n\) items taken \(k\) at a time. Luckily, when \(n = 2\), using the basic formula: \((a + b)^2 = a^2 + 2ab + b^2\), it becomes a lot easier.

This simplified form made it easy for us to expand \((3x + 2y)^2\) into \(9x^2 + 12xy + 4y^2\). The concepts introduced by the binomial theorem are used extensively in probability, statistics, and other areas of mathematics.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is a unifying thread of almost all mathematics. In our exercise, we take advantage of our algebraic knowledge to manipulate the symbols within the binomial to achieve a polynomial expansion.

Polynomials are expressions that consist of variables and coefficients, typically connected by operations of addition, subtraction, and multiplication. Here, we transformed a basic binomial expression \((3x + 2y)^2\) into a polynomial format. This requires not only the binomial theorem but also foundational algebra skills to plug numbers correctly, compute products, and combine like terms.

In algebra, the ability to perform these manipulations accurately helps us solve equations and transform expressions into their most useful forms. This exercise illustrates how employing standard algebraic procedures can simplify complex expressions into straightforward ones.

Quadratic Expansion

Quadratic expansion is a specific application of the more general binomial expansion and focuses on squares of binomials, like in our problem. The idea here is to convert a squared binomial such as \((a + b)^2\), into a trinomial. Remember the formula that guides us: \((a + b)^2 = a^2 + 2ab + b^2\).
In this exercise, we approached the quadratic expansion by identifying the components of our binomial: \(a = 3x\) and \(b = 2y\). We then used these components to individually calculate each term:

• First, of course, is \(a^2 = (3x)^2 = 9x^2\).
• The middle term, \(2ab = 2(3x)(2y) = 12xy\), combines both parts of the binomial.
• Finally, \(b^2 = (2y)^2 = 4y^2\) completes the expansion.

Adding these results provides the fully expanded quadratic expression \(9x^2 + 12xy + 4y^2\). By mastering quadratic expansion of binomials, you gain the ability to transform squared expressions into more manageable polynomials, which is an essential technique in algebra.