/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 Find each product. $$ \left(... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 34

Find each product. $$ \left(9 y^{2}-2\right)\left(9 y^{2}+2\right) $$

Short Answer

Expert verified
81y^4 - 4

Step by step solution

01

- Identify the expression format

The given expression is in the format of \((a - b)(a + b)\), which can be simplified using the difference of squares formula: \[ (a - b)(a + b) = a^2 - b^2 \]
02

- Determine the values of a and b

In this problem, \(a\) corresponds to \(9y^2\) and \(b\) corresponds to 2.
03

- Apply the difference of squares formula

Using the formula \[ (a - b)(a + b) = a^2 - b^2 \], substitute \(a = 9y^2\) and \(b = 2\).
04

- Calculate the squares

Calculate \(a^2\): \[ a^2 = (9y^2)^2 = 81y^4 \] and \(b^2\): \[ b^2 = 2^2 = 4 \]
05

- Substitute and simplify

Substitute the squared terms back into the formula: \[ (9y^2)^2 - 2^2 = 81y^4 - 4 \] Therefore, the product is: \[ 81y^4 - 4 \]

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Key Concepts

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

Polynomial Multiplication
Understanding polynomial multiplication is a key skill in algebra. It involves multiplying polynomials by combining like terms. Here, we work with two binomials \( (9y^2 - 2)(9y^2 + 2) \). Multiplying these requires recognizing the pattern as the difference of squares. Polynomial multiplication involves each term of one polynomial multiplying each term of the other. For binomials, use the FOIL method or patterns like the difference of squares. Recognize patterns for easier manipulation and solve efficiently.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In this exercise, \( 9y^2 - 2 \) and \( 9y^2 + 2 \) are our algebraic expressions. It's essential to understand how these expressions interact during multiplication. Recognize that when expressions are multiplied, you distribute each term to all terms in the other expression. In the case of difference of squares, using the specific formula simplifies the process. This approach reduces complex steps, making calculations more straightforward.
Factoring
Factoring is breaking down algebraic expressions into simpler multipliers. The difference of squares formula is a factoring tool: \( (a-b)(a+b) = a^2 - b^2 \). In our example, \( 9y^2 - 2 \) and \( 9y^2 + 2 \) can be factored using this formula. Factoring simplifies expressions and solving equations. Apply \( a = 9y^2 \) and \( b = 2 \), then solve \[ (9y^2)^2 - 2^2 = 81y^4 - 4 \]. It turns complex expressions into manageable forms for easier solving.

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Most popular questions from this chapter

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