Problem 74 - College Algebra Essentials

Question

In Exercises 67–82, find each product. $$(9 x+7 y)^{2}$$

Step-by-Step Solution

Answer

\((9x + 7y)^2 = 81x^2 + 126xy + 49y^2\)

Step 1: Identify the Components of the Binomial

In the expression \((9x + 7y)^2\), identify \(a\) and \(b\). In this case, \(a = 9x\) and \(b = 7y\).

Step 2: Apply the Binomial Square Formula

Apply the formula \((a + b)^2 = a^2 + 2ab + b^2\). Substitute \(a\) and \(b\) in the formula to get \((9x + 7y)^2 = (9x)^2 + 2 * 9x * 7y + (7y)^2\).

Step 3: Simplify the Expression

Calculate the squares and the product terms in the expression: \((9x)^2 = 81x^2\), \(2 * 9x * 7y = 126xy\) and \((7y)^2 = 49y^2\). Therefore, \((9x + 7y)^2 = 81x^2 + 126xy + 49y^2\)

Key Concepts

Understanding Algebra in Binomial Expansion

Algebra is a branch of mathematics that uses symbols, often letters, to represent numbers in equations and expressions. This allows us to perform operations in a more general way. In the problem, algebra is used to expand the binomial expression \((9x + 7y)^2\). This involves manipulating symbols, so we can simplify the expression using basic algebraic principles. Knowing how to handle variables like \(x\) and \(y\) is key because it helps us substitute and simplify more complex real-world problems. Using roots, powers, and coefficients is fundamental in algebra, as they appear frequently when dealing with polynomial expressions, like in our exercise.

The Role of the Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that describes the algebraic expansion of powers of a binomial. A binomial is an expression with two terms, like \((9x + 7y)\). In our exercise, we use the case of squaring a binomial. The standard formula for squaring a binomial is:

• \((a + b)^2 = a^2 + 2ab + b^2\)

When applying this to \((9x + 7y)^2\), we identify \(a = 9x\) and \(b = 7y\). Plugging these into the binomial expansion formula, each component involves simplifying these terms. The squared terms \((9x)^2\) and \((7y)^2\), and the middle term based on \(2ab\), give us the full expanded form. Using the binomial theorem simplifies polynomial expansions, making it easy to handle higher-power expressions.

Polynomials and Their Expansion

Polynomials are expressions made up of variables and coefficients, involving operations of addition, subtraction, and multiplication. Each part of the polynomial is called a term.In our expanded expression

• \((9x + 7y)^2 = 81x^2 + 126xy + 49y^2\)

we get a polynomial with three terms. - \(81x^2\) is the square of the first term.- \(49y^2\) is the square of the second term.- \(126xy\) is a cross product of the terms.Polynomials are versatile and can represent a variety of functions. Understanding their expansion helps identify each term separately, allowing for deeper insights into algebraic operations and simplifying problems across fields such as engineering, physics, and economics.