Problem 78 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (4 x-9 y)^{2} $$

Step-by-Step Solution

Answer

The square of \((4x - 9y)\) is \(16x^2 - 72xy + 81y^2\).

Step 1: Understand the Expression

The given problem is \((4x - 9y)^2\). This means you need to square the expression \(4x - 9y\), which is equivalent to multiplying \((4x - 9y)\) by itself.

Step 2: Apply the Binomial Square Formula

Recall the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 4x\) and \(b = 9y\). Substitute these into the formula.

Step 3: Calculate Each Part of the Formula

First, calculate \(a^2 = (4x)^2 = 16x^2\). Next, calculate \(b^2 = (9y)^2 = 81y^2\). Finally, calculate \(2ab = 2 \cdot 4x \cdot 9y = 72xy\).

Step 4: Substitute Back into the Formula

Substitute the calculated parts back into the formula: \((4x - 9y)^2 = 16x^2 - 72xy + 81y^2\).

Step 5: Simplify and Write Final Answer

The expression is already simplified, so the final answer is: \(16x^2 - 72xy + 81y^2\).

Key Concepts

Binomial Expansion

The binomial expansion is a powerful algebraic technique that simplifies the process of expanding the power of a binomial. When you see an expression like \((a - b)^2\), your task is to expand it to reveal its full polynomial form. By employing the square of a binomial formula \((a - b)^2 = a^2 - 2ab + b^2\), you can easily transform the expression. This formula serves as an essential tool in mathematics because it allows you to quickly and accurately expand the square of any binomial expression. For example, to expand \((4x - 9y)^2\), you substitute \(a = 4x\) and \(b = 9y\) into the formula, simplifying your work and ensuring correctness.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a value or relationship. Understanding the structure of an algebraic expression is key to solving problems. In the expression \((4x - 9y)^2\), the variables \(x\) and \(y\) can take on different values, which means that the expression can represent different outcomes. By squaring the expression using binomial expansion, you can express it as a polynomial, illustrating its complete form. Each term in the expanded polynomial, such as \(16x^2\), \(-72xy\), and \(81y^2\), represents part of the solution, showing how each variable contributes to the expression's overall value. Understanding this concept helps you break down complex algebraic expressions and solve them step by step.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials together. It's an essential process in algebra that requires you to carefully expand each part. The problem \((4x - 9y)^2\) demonstrates a specific type of polynomial multiplication called the square of a binomial. Here, you multiply the binomial \((4x - 9y)\) by itself. Using the binomial expansion formula makes this process efficient and straightforward. You calculate each term: \(16x^2\) from \((4x)^2\), \(-72xy\) from \(-2 \cdot 4x \cdot 9y\), and \(81y^2\) from \((9y)^2\). Each of these products contributes to the final polynomial form, displaying the results of the polynomial multiplication. By mastering polynomial multiplication, you'll become adept at simplifying and solving complex expressions.