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Chapter 4: Problem 38

Find each product. $$ \left(7 x+\frac{3}{7}\right)\left(7 x-\frac{3}{7}\right) $$

Short Answer

Expert verified
49x^2 - \frac{9}{49}

Step by step solution

01

Identify the Structure

Recognize that the given expression is in the form \((a + b)(a - b)\), which matches the difference of squares formula.
02

Apply the Difference of Squares Formula

Use the formula: \[ (a + b)(a - b) = a^2 - b^2 \]. Here, \ a = 7x \ and \ b = \frac{3}{7} \.
03

Square the First Term

Calculate \( a^2 \): \[ (7x)^2 = 49x^2 \].
04

Square the Second Term

Calculate \( b^2 \): \[ \left(\frac{3}{7}\right)^2 = \frac{9}{49} \].
05

Subtract the Squares

Combine the results using the difference of squares formula: \[ 49x^2 - \frac{9}{49} \].

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and mathematical operations. They form the core components of algebra problems. In our exercise, the expression \( (7x + \frac{3}{7})(7x - \frac{3}{7}) \) consists of terms including a variable \( x \) and constants. By recognizing and manipulating such expressions, we can simplify and solve complex problems. Understanding how to break down and restructure these expressions is crucial for tackling algebraic challenges.
Multiplication
Multiplication of algebraic expressions involves distributing each term in the first expression by each term in the second expression. In specific cases like this exercise, where we use the difference of squares formula, the multiplication process gets streamlined. Instead of multiplying each term traditionally, we can use: \[ (a + b)(a - b) = a^2 - b^2 \]. This helps us quickly simplify the expression without going through multiple steps. It emphasizes why knowing key formulas can save time and reduce errors in algebra.
Polynomial Identities
Polynomial identities are widely used in algebra to simplify expressions and solve equations efficiently. In our example, recognizing the identity \( (a + b)(a - b) = a^2 - b^2 \) allows for quick simplification: \[ (7x + \frac{3}{7})(7x - \frac{3}{7}) = (7x)^2 - (\frac{3}{7})^2 = 49x^2 - \frac{9}{49} \]. This identity belongs to a set of polynomial identities that includes other formulas like the square of a sum, square of a difference, and sum/difference of cubes. Mastery of these identities is key for excelling in algebra, as they provide powerful tools for problem-solving.

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Most popular questions from this chapter

Each statement contains a number in boldface italics. Write the number in scientific notation. In 2007 , assets of the insured commercial banks in the United States totaled about 13,039,000,000,000 dollars. (Source: U.S. Federal Deposit Insurance Corporation.)

Find each product. $$ (3 x+4 y)(3 x-4 y) $$

Use scientific notation to calculate the answer to each problem. Write answers in scientific notation. $$ \frac{0.000015(42,000,000)}{0.000009(0.000005)} $$

Use scientific notation to calculate the answer to each problem. Write answers in scientific notation. $$ \frac{3,400,000,000(0.000075)}{0.00025} $$

Find each product. $$ \left(9 y+\frac{2}{3}\right)\left(9 y-\frac{2}{3}\right) $$
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