Problem 117 - Introductory Algebra for College Students

Question

Perform the indicated operation. $$(4 x+5 y)^{2}$$

Step-by-Step Solution

Answer

The expanded expression of \((4x + 5y)^2\) is \(16x^2 + 40xy + 25y^2\).

Step 1: Expanding the Square

In the formula \((a+b)^2 = a^2 + 2ab + b^2\), identify \(a\) as \(4x\) and \(b\) as \(5y\). Substitute these values into the square of a binomial formula: \((4x)^2 + 2(4x)(5y) + (5y)^2\).

Step 2: Solving the Square Terms

Next, solve each term individually. Square \(4x\) to get \(16x^2\), multiply \(2(4x)(5y)\) to get \(40xy\), and square \(5y\) to get \(25y^2\). The simplified expression is therefore, \(16x^2 + 40xy + 25y^2\).

Key Concepts

Expanding Polynomials

Expanding polynomials is a fundamental skill in algebra which involves simplifying expressions to remove parentheses. It's essential when dealing with equations, functions, and various types of algebraic problems. When you see a polynomial, like the square of a binomial ewline\((a+b)^2\), ewline the process of expanding involves distributing each term in the first binomial across each term in the second binomial.

In our example from the exercise, ewline\((4x + 5y)^2\), ewline we apply this technique by multiplying each term in the binomial by every other term, including itself, as per the formula ewline\(a^2 + 2ab + b^2\).ewline Here, ewline\(a = 4x\) ewline and ewline\(b = 5y\).ewline By distributing we find that ewline\((4x)^2 = 16x^2\), ewline\(2(4x)(5y) = 40xy\), ewline and ewline\((5y)^2 = 25y^2\).ewline Consequently, the expanded form becomes ewline\(16x^2 + 40xy + 25y^2\).ewline It's crucial to working methodically to avoid missing any terms or introducing errors in the calculation.

Remember, when expanding-polynomials, always pay attention to the signs (positive or negative) of the terms you're multiplying, as this can greatly affect the outcome.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In algebra, expressing relationships between variables and constants succinctly is key to solving equations and understanding functions. A square of a binomial is a specific type of algebraic expression which consists of a binomial (a two-termed expression) raised to the second power.

The structure of these expressions reflects the mathematical relationships they represent. For ewline\((4x + 5y)^2\), ewline we are looking at two algebraic expressions ewline\(4x\) ewline and ewline\(5y\) ewline and considering the square of their sum. The key to working with such expressions is understanding the order of operations and the properties of exponents and arithmetic operations. Using the distributive property allows us to multiply these expressions together.

For instance, an important detail to note is that when squaring a term, you square both the coefficient (number) and the variable independently. This is why we get from ewline\((4x)^2\)ewline to ewline\(16x^2\), ewline and from ewline\((5y)^2\) ewline to ewline\(25y^2\).ewline The multiplication of terms also follows the commutative property of multiplication, simplifying the process of combining like terms.

Binomial Theorem

The binomial theorem is a powerful tool in algebra that provides a quick way to expand binomials raised to a power. Its application goes beyond just squaring to dealing with any exponent. This theorem generalizes the pattern of coefficients that arise when a binomial like ewline\((a+b)^n\) ewline is expanded. For the square of a binomial, which is the special case where ewline\(n = 2\), ewline the binomial theorem tells us that we will have three terms, one from squaring ewline\(a\), ewline one from doubling the product of ewline\(a\) ewline and ewline\(b\), ewline and one from squaring ewline\(b\).ewline The coefficients (1, 2, 1) are also the second row of Pascal's Triangle, which is intimately connected with the binomial theorem.

In essence, the theorem helps us predict and verify the results of expanding polynomials without going through the multiplication for every term, especially useful for higher powers. When applied to our example ewline\((4x + 5y)^2\), ewline it immediately indicates the three terms in the expanded form. Recognizing these patterns ensures accuracy, saving time and effort, particularly for more complex binomials.