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Chapter 6: Problem 90

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

Short Answer

Expert verified
81 - y^2

Step by step solution

01

Identify the Binomials

Recognize that you are multiplying two binomials in the form \(a-b\) and \(a+b\). Here, \(a=9\) and \(b=y\).
02

Apply the Difference of Squares Formula

Use the formula for the difference of squares, which states that \( (a-b)(a+b) = a^2 - b^2 \).
03

Substitute the Values

Substitute \(a=9\) and \(b=y\) into the formula: \( 9^2 - y^2 \).
04

Simplify

Calculate \( 9^2 \) to get 81. Therefore, the expression simplifies to \( 81 - y^2\).

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Key Concepts

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

Understanding Binomials
In algebra, a binomial is a polynomial with exactly two terms. These terms are usually connected by a plus or minus sign. For example, in the exercise given, the binomials are \(9-y\) and \(9+y\). Each binomial has two parts: a constant (which is 9 in both) and a variable (represented by \(y\)).
The structure of binomials often makes them useful in various algebraic operations, such as factoring and expanding expressions.
When working with binomials, identifying the terms is the first step in solving algebraic problems. Knowing the components allows us to apply specific formulas to simplify or transform the expressions effectively.
Difference of Squares
The concept of difference of squares is a specific pattern used in algebra to simplify the multiplication of certain binomials.
The difference of squares formula states that: \((a - b)(a + b) = a^2 - b^2\).
This formula is handy when you have binomials structured as one term subtracted from another and one term added to another, like \(9 - y\) and \(9 + y\).
By using this formula, the product of the binomials \(9 - y\) and \(9 + y\) simplifies directly to \(9^2 - y^2\), eliminating the need for further distribution and combination of terms.
Simplification
Simplifying an algebraic expression means reducing it to its most straightforward form.
In this exercise, after applying the difference of squares formula to the binomials, we get an expression: \(9^2 - y^2\).
Simplification requires calculating any constants, so here, we compute \(9^2\) which equals 81.
Thus, the expression \(81 - y^2\) is the simplest form of the original problem \((9-y)(9+y)\).
Simplification is crucial in algebra as it makes expressions cleaner and easier to interpret or solve in further mathematical problems.

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