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Chapter 9: Problem 103

Multiply. $$(4 x-9 y)(4 x+9 y)$$

Short Answer

Expert verified
The result of the multiplication \((4 x - 9 y)(4 x + 9 y)\) is \(16x^2 - 81y^2\).

Step by step solution

01

Identify 'a' and 'b' in the given expression

In the given expression \((4 x-9 y)(4 x+9 y)\), \(a = 4x\) and \(b = 9y\).
02

Apply the Difference of Squares formula

Now we need to apply the difference of squares formula \((a+b)(a-b) = a^2 - b^2\). In our case, it will be like \((4x)^2 - (9y)^2\).
03

Simplify the Expression

Now, simplify the expression by squaring the terms. We get \(16x^2 - 81y^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The concept of the difference of squares is a handy tool in algebra for multiplying certain types of polynomials. When you see an expression in the form of
  • difference of squares: \((a + b)(a - b)\), remember it transforms into the simpler form \(a^2 - b^2\).
This means, instead of multiplying each term separately, we can directly square the individual terms in parentheses and subtract.

In the example, the expression is \((4x - 9y)(4x + 9y)\), which fits perfectly into the difference of squares rule. Here, \(a = 4x\) and \(b = 9y\). Applying the formula gives us; "

  • \((4x)^2 - (9y)^2\)
Always check if your algebraic expression can be modified using this useful property to make multiplication much easier.
Multiplying Binomials
Multiplying binomials is a standard operation in algebra where two expressions, each containing two terms, are multiplied against each other. The difference of squares is a special case, but the general approach is still essential to understand.

For two binomials: \((a+b)\) and \((a-b)\), we usually utilize the *FOIL* method.
  • The First terms: \(a*a\)
  • The Outer terms: \(a*b\)
  • The Inner terms: \(b*a\)
  • The Last terms: \(b*b\)
FOIL is great for two general binomials but doesn't apply directly to the difference of squares scenario where middle terms cancel each other out. Thus resulting in only:
  • \(a^2-b^2\)
So, in difference of squares, you avoid full expansion, making it quicker and easier to compute.
Algebraic Expressions
An algebraic expression is a vital concept in mathematics, representing a combination of numbers, variables, and operating symbols. These expressions can include polynomials such as our given example \((4x - 9y)(4x + 9y)\).

Expressions can be simplified to make operations and solutions more straightforward. This often involves recognizing patterns, such as the difference of squares, where expressions can be evaluated more efficiently.
  • Terms: Parts of the expression separated by addition or subtraction symbols. In the example, 'terms' are \(4x\) and \(-9y\).
  • Variables: Symbols for numbers we don't know yet or those that can vary. Here, \(x\) and \(y\) are variables.
  • Coefficients: Constants used to multiply the variables. For instance, \(4\) is the coefficient of \(x\).
Understanding these parts helps in simplifying and manipulating expressions efficiently, making it easier to handle algebraic operations.

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Most popular questions from this chapter

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