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Chapter 9: Problem 100

Multiply. $$(3 x-7)(3 x+7)$$

Short Answer

Expert verified
The answer is \(9x^2 - 49\).

Step by step solution

01

Identifying the binomials

Identify the two binomials: \(3x - 7\) and \(3x + 7\). Here, \(a = 3x\) and \(b = 7\).
02

Applying the formula

Apply the formula for the difference of squares: \((a-b)(a+b) = a^2 - b^2\). Substitute \(a\) and \(b\) into the formula: \((3x - 7)(3x + 7) = (3x)^2 - 7^2\).
03

Simplifying the expression

Simplify the expression : \((3x)^2 - 7^2 = 9x^2 - 49\).

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Key Concepts

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

Multiplying Binomials
When you encounter an expression like \((3x - 7)(3x + 7)\), you are working with binomials. A binomial is simply an algebraic expression with two terms, such as \(3x - 7\) and \(3x + 7\). The goal is to multiply these binomials to form a single expression.

This type of multiplication often uses a specific pattern called the "Difference of Squares." According to this pattern, multiplying the binomials looks like this:

  • Acknowledge each binomial: \(3x - 7\) and \(3x + 7\).
  • Recognize this as a special case because both binomials have the same terms, just arranged oppositely.
For binomials with this arrangement, you apply the difference of squares formula:

\[(a - b)(a + b) = a^2 - b^2\]

This formula simplifies the multiplication process and results in \((3x)^2 - 7^2\), without needing to multiply each term individually.
Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. In our case, the expressions \(3x - 7\) and \(3x + 7\) are binomials made of:
  • Variables: \(x\) which represents an unknown value.
  • Constants: \(-7\) and \(+7\) are fixed values.
  • Coefficients: \(3\) is the number multiplying the variable \(x\).
Understanding these elements helps in manipulating and simplifying algebraic expressions effectively.

When multiplying binomials like these, you use their structure to apply specific algebraic patterns or identities, such as the difference of squares. This is crucial for simplifying the expressions into a more manageable form.

By recognizing the pattern, you're able to skip several steps that would otherwise require multiplying and combining like terms separately.
Simplifying Expressions
Once you have applied the difference of squares formula, you shift to simplifying the expression. Simplification means reducing the expression to its simplest form. In mathematical terms, this means working with the result \((3x)^2 - 7^2\).

To simplify:
  • Calculate \((3x)^2\):
    \(3x \times 3x = 9x^2\)
  • Calculate \(7^2\):
    \(7 imes 7 = 49\)
With these results, replace in the difference of squares formula:
\[9x^2 - 49\]
This is our simplified expression, which is the product of the binomials \(3x - 7\) and \(3x + 7\).

The simplification involves reducing the expression without changing its value, resulting in a more concise and easily interpretable form. Understanding this simplification process ensures accuracy in solving algebraic problems.

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