Problem 24 - Elementary Algebra

Question

For the following problems, find the products. $$ (3 x+5)^{2} $$

Step-by-Step Solution

Answer

Based on the given expression \((3x+5)^2\), we used the binomial theorem to find the product. After identifying the values of a (\(3x\)), b (\(5\)), and n (\(2\)), we expanded the expression using the theorem. The final product of the expression is \(9x^2 + 30x + 25\).

Step 1: Identify the values of a, b, and n

We have a binomial expression, and so we identify the values of \(a\), \(b\), and \(n\) as follows: - \(a = 3x\) - \(b = 5\) - \(n = 2\)

Step 2: Apply the binomial theorem

Using the binomial theorem mentioned in the analysis, we can expand the given expression. We only need to calculate the first three terms since \(n=2\). - \(\binom{2}{0} a^2 b^0 = \binom{2}{0} (3x)^2 (1) = 1 \cdot 9x^2 = 9x^2\) - \(\binom{2}{1} a^1 b^1 = \binom{2}{1} (3x)^1 (5) = 2 \cdot 15x = 30x\) - \(\binom{2}{2} a^0 b^2 = \binom{2}{2} (1) (5)^2 = 1 \cdot 25 = 25\)

Step 3: Combine the terms and simplify

Now we can combine the terms we calculated in step 2 to expand the given expression: $$ (3x + 5)^2 = 9x^2 + 30x + 25 $$ So, the product of the given expression is: $$ (3x + 5)^2 = 9x^2 + 30x + 25 $$

Key Concepts

Binomial Expression

Understanding the basics of binomial expressions is vital in algebra. A binomial is a polynomial which is the sum of exactly two terms, and each term is called a monomial. In the exercise \( (3x + 5)^2 \), the binomial expression consists of two monomials: \(3x\) and \(5\).

When we look at the operation involved, raising a binomial to a power, like the square in this exercise, we're dealing with repeated multiplication of the binomial by itself. Here's where the binomial theorem comes into play as a powerful shortcut for expanding binomial expressions raised to positive integer powers, without conducting the multiplication manually. It states that \( (a + b)^n \) can be expanded into the sum of terms involving the coefficients from Pascal's triangle, \(a\)'s powers decreasing from \(n\) to \(0\), and \(b\)'s powers increasing from \(0\) to \(n\).

This theorem not only simplifies the process but also ensures accuracy in the expansion of binomial expressions.

Algebraic Products

The concept of algebraic products revolves around multiplying algebraic expressions together. It is a fundamental operation in algebra that leads to the creation of new expressions. In the context of the given exercise, the multiplication is between two identical binomial expressions, \( (3x + 5) \), which is a special case known as a square of a binomial.

The process can be visualized in two ways: either by using the distributive property—commonly known by the mnemonic FOIL (First, Outer, Inner, Last)—to multiply each term in one binomial by each term in the other, or by applying the binomial theorem as seen in the exercise solution.

By addressing algebraic products through the lens of the binomial theorem, we cut down on the number of individual multiplications and add a layer of clarity to the process, especially as the power increases beyond squaring.

Polynomial Simplification

After using the binomial theorem to expand a binomial expression, it's time to transition into the realm of polynomial simplification. This involves combining like terms and arranging them, if needed, in descending powers of the variables involved. Simplification makes the expression neater and often easier to use in subsequent calculations or to analyze its properties.

In the completed exercise, each term of the expanded polynomial \(9x^2 + 30x + 25\) is already simplified, as there are no like terms to combine. However, it's important to remember that simplification may be more involved in other contexts, requiring careful attention to combining like terms correctly. Ultimately, polynomial simplification is what ties the process together, groomed from an expanded binomial expression into a clear, concise polynomial ready for further use.