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Chapter 4: Problem 35

Find each product. $$ \left(\frac{3}{4}-x\right)\left(\frac{3}{4}+x\right) $$

Short Answer

Expert verified
The product is \( \frac{9}{16} - x^2\).

Step by step solution

01

Identify the terms

Identify the terms in both binomials. Here, the binomials are \(\frac{3}{4}-x\) and \( \frac{3}{4}+x\).
02

Apply the difference of squares formula

Apply the formula for the product of the sum and difference of the same two terms. The formula is \((a-b)(a+b) = a^2 - b^2\).
03

Substitute the values

Substitute \(a \) with \(\frac{3}{4}\) and \(b \) with \(x \) into the difference of squares formula. This gives \(\frac{3}{4}^2 - x^2\).
04

Square the fraction

Square \(\frac{3}{4}\) to get \(\frac{9}{16}\). The expression now becomes \(\frac{9}{16} - x^2\).

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Key Concepts

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

Binomials
A binomial is an algebraic expression that contains exactly two terms. These terms can be numbers, variables, or a combination of both. In the given problem, the binomials are \(\frac{3}{4} - x\) and \(\frac{3}{4} + x\). Each binomial has one term with a fraction and one term with a variable.

Binomials are commonly used in algebra to represent expressions that can undergo various operations, such as addition, subtraction, multiplication, and division. Understanding how to work with binomials is essential for solving more complex algebraic problems.
Product of Binomials
The product of binomials involves multiplying two binomial expressions. In this exercise, we are multiplying \(\frac{3}{4} - x\) and \(\frac{3}{4} + x\). This type of multiplication can often simplify to a more manageable form.

When multiplying binomials, we use the distributive property to ensure that each term in the first binomial is multiplied by each term in the second binomial. However, in some cases, special algebraic formulas can simplify the process, such as the difference of squares formula used in this problem.
Algebraic Identities
Algebraic identities are formulas that hold true for all values of the variables within them. These identities simplify complex expressions and make algebraic manipulation quicker and more intuitive.

One of the most common algebraic identities is the difference of squares, formulated as \((a-b)(a+b) = a^2 - b^2\). This identity is particularly helpful when dealing with binomials like \(\frac{3}{4} - x\) and \(\frac{3}{4} + x\). It eliminates the middle terms and directly gives us a simpler expression, \(\frac{3}{4}^2 - x^2\).
Squared Terms
A squared term is a term that has been raised to the power of two. In this exercise, we encounter squared terms when applying the difference of squares formula.

For the binomials given, we substitute \(\frac{3}{4}\) for \(a\) and \x\ for \(b\) in the formula \((a-b)(a+b) = a^2 - b^2\). The squared terms, \(\frac{3}{4}^2\) and \(x^2\), simplify to \(\frac{9}{16}\) and \(x^2\), respectively. Understanding squared terms is crucial as they frequently appear in algebra and higher-level mathematics.

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Most popular questions from this chapter

Explain how to multiply a number by a positive power of 10. Then explain how to multiply a number by a negative power of 10.

Each statement contains a number in boldface italics. Write the number in scientific notation. At the end of 2007 , the total number of cellular telephone subscriptions in the world reached about \(3,305,000,000 .\) (Source: International Telecommunications Union.)

Evaluate each expression for \(x=3 .\) See Sections 1.3 and 1.6. $$ 3 x^{2}-3 x+4 $$

Use scientific notation to calculate the answer to each problem. See Examples 3-5. During the \(2007-2008\) season, Broadway shows grossed a total of \(9.38 \times 10^{8}\) dollars. Total attendance for the season was \(1.23 \times 10^{7} .\) What was the average ticket price for a Broadway show? (Source: The Broadway League.)

Find each product. $$ -3 k\left(8 k^{2}-12 k+2\right) $$
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