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Chapter 4: Problem 32

Find each product. $$ (7-2 x)(7+2 x) $$

Short Answer

Expert verified
49 - 4x^2

Step by step solution

01

- Recognize the Pattern

Identify that the expression (7-2x)(7+2x) is in the form of a difference of squares: (a-b)(a+b) = a^2 - b^2.
02

- Define Variables

Assign the variables: a = 7 and b = 2x.
03

- Apply the Difference of Squares Formula

Substitute the variables into the formula (a^2 - b^2): a = 7 and b = 2x. Then, it becomes: (7)^2 - (2x)^2.
04

- Simplify

Calculate the squares: (7)^2 = 49 and (2x)^2 = 4x^2. Subtract these results: 49 - 4x^2.

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Key Concepts

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

Polynomial Multiplication
Polynomial multiplication involves multiplying each term in one polynomial by each term in another polynomial. This process can be simplified using specific algebraic patterns. For instance, in our exercise, we use a special case called the 'difference of squares'.
The difference of squares formula is \((a - b)(a + b) = a^2 - b^2\). This helps in quickly multiplying two binomials and getting a simplified result.
Understanding this pattern allows you to skip tedious multiplication steps and directly simplify expressions.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations (like addition, subtraction, multiplication, and division).
In our exercise, the expression \(7 - 2x\) and \(7 + 2x\) are examples of binomial expressions.
These expressions follow fundamental algebraic rules and patterns. Recognizing patterns such as the difference of squares can drastically simplify complex expressions.
By assigning variables (like \(a = 7\) and \(b = 2x\)), complex expressions can become more manageable.
Simplifying Expressions
Simplifying expressions means to reduce them to their simplest form. This often involves combining like terms or using algebraic identities. For our exercise, simplifying \((7 - 2x)(7 + 2x)\) started with recognizing it as a difference of squares.
This allowed us to apply the formula \(a^2 - b^2\), making the expression easier to handle. After substituting the values: \(a = 7\) and \(b = 2x\), we simplified \( (7)^2 - (2x)^2 = 49 - 4x^2\).
Always ensure you simplify each part of the expression separately before combining them for the final result.

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Most popular questions from this chapter

Each statement contains a number in boldface italic type. If the number is in scientific notation, write it in standard notation. If the number is not in scientific notation, write it as such. See Examples 1 and \(2 .\) 42\. There are 13 red balls and 39 black balls in a box. Mix them up and draw 13 out one at a time without returning any ball \(\ldots\) the probability that the 13 drawings each will produce a red ball is \(\ldots 1.6 \times 10^{-12}\). (Data from Weaver, W., Lady Luck.)

If a negative number is raised to an even power, the result is positive. For example, $$(-3)^{4}=(-3)(-3)(-3)(-3)=+81-\text { that is, } 81$$ If a negative number is raised to an odd power, the result is negative. For example, $$(-3)^{5}=(-3)(-3)(-3)(-3)(-3)=-243$$ Without actually performing the computation, state whether the power is positive or negative. (a) \((-4)^{3}\) (b) \((-4)^{4}\) (c) \((-4)^{12}\) (d) \((-4)^{13}\)

A student simplified the expression \(\left(10^{2}\right)^{3}\) as \(1000^{6}\). This is incorrect. WHAT WENT WRONG? Simplify correctly.

Find each product. $$ (2 m+5)(2 m-5) $$

Combining like terms in \(-3 x y-2 x y+5 x y\) gives _____________.
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