Problem 52 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(9 x-2 y)(9 x+2 y)$$

Step-by-Step Solution

Answer

The product is \(81x^2 - 4y^2\).

Step 1: Recognize the pattern

The expression \((9x-2y)(9x+2y)\) fits the pattern of a difference of squares, \((a-b)(a+b) = a^2 - b^2\). In this expression, \(a = 9x\) and \(b = 2y\).

Step 2: Apply the difference of squares formula

Now use the difference of squares formula to find the product. Substitute \(a = 9x\) and \(b = 2y\) into the formula, which gives us \(a^2 - b^2 = (9x)^2 - (2y)^2\).

Step 3: Calculate the squares

First, square \(9x\) to get \((9x)^2 = 81x^2\). Then, square \(2y\) to get \((2y)^2 = 4y^2\).

Step 4: Subtract the squares

Subtract \(4y^2\) from \(81x^2\) using the difference of squares result: \(81x^2 - 4y^2\).

Step 5: Write the final result

Combine all the results to get the final product expression: \(81x^2 - 4y^2\).

Key Concepts

Difference of Squares

The difference of squares is a special mathematical pattern that makes certain calculations straightforward and quicker. This pattern occurs when you multiply two binomials that are identical except for the sign between their terms. In formula terms, it looks like \[(a-b)(a+b) = a^2 - b^2\].When you recognize

• one term as a subtraction
• the other as an addition

you can apply this pattern. For the expression \((9x-2y)(9x+2y)\),we let \(a = 9x\) and \(b = 2y\). Quickly identifying that this fits the difference of squares pattern lets us solve it in a snap! The result will directly be \(a^2 - b^2 = (9x)^2 - (2y)^2\).
This technique is an efficient shortcut simplifying multiplication into subtraction of squares.

Binomial Multiplication

Binomial multiplication is a fundamental concept that involves multiplying two expressions, each containing two terms. These expressions are known as binomials. The formula typically used in our scenario is:\[(a+b)(a-b) = a^2 - b^2\],but more generally it can be expanded using the distributive property (also known as FOIL: First, Outer, Inner, Last).
For any two binomials, say \((a+c)(b+d)\),you would multiply:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms in the product.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

If the binomials \((a-b)(a+b)\)fit the difference of squares as in this exercise, you simply apply the special formula for a quicker solution, derived from this very property. No need to actually compute each pair separately in the specific case.

Special Product Patterns

When dealing with binomials, recognizing special product patterns speeds up the multiplication process significantly. Special patterns like these are pre-defined ways to simplify the multiplication of certain types of binomials.
The two critical types you will frequently encounter are:

• **Perfect Square Trinomials**: Products that follow from squaring a binomial, e.g., \((a+b)^2=a^2+2ab+b^2\)
• **Difference of Squares**: As previously mentioned, \((a-b)(a+b) = a^2 - b^2\)

Recognizing these patterns helps solve such expressions swiftly without much ado. For example, identifying the problem \((9x-2y)(9x+2y)\)as a difference of squares allowed us to directly apply the formula \((9x)^2 - (2y)^2 = 81x^2 - 4y^2\).Keep these patterns handy and use them wherever applicable to make your calculations razor-sharp and time-efficient.