Problem 83 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$(4 x-5)(4 x+5)$$

Step-by-Step Solution

Answer

The product of \( (4x - 5)(4x + 5) \) is \( 16x^2 - 25 \).

Step 1: Identify the terms in the binomial

In the binomial multiplication \( (4x - 5)(4x + 5) \), the terms can be identified as \( a = 4x \) and \( b = 5 \).

Step 2: Apply the formula

Applying the formula \( (a - b)(a + b) = a^2 - b^2 \), the product of the binomials will be the difference of squares of \( a \) and \( b \), i.e., \( (4x)^2 - (5)^2 \).

Step 3: Solve for the result

The square of \( 4x \) is \( 16x^2 \) and the square of \( 5 \) is \( 25 \). So, the difference between these squares is \( 16x^2 - 25 \).

Key Concepts

Difference of Squares

The difference of squares is a straightforward algebraic pattern. It shows up often when you multiply two binomials with the form \((a - b)(a + b)\). This pattern is beneficial as it simplifies the process of multiplying these expressions.
When you apply this formula, \((a - b)(a + b) = a^2 - b^2\), it means you only need to find the square of each term both once.

• In our example, \((4x - 5)(4x + 5)\), the term \(a\) is \(4x\) and \(b\) is \(5\).
• The square of \(4x\) is \(16x^2\)
• The square of \(5\) is \(25\).

The difference of these squares is simply calculated: \(16x^2 - 25\).
This method bypasses "FOIL" and other multiplication techniques, providing a quicker way to the answer without intermediate steps.

Standard Form

In polynomial expressions, standard form is a way of writing terms with descending powers. This helps in organizing and simplifying polynomials for further operations or to easily identify the degree.
For any polynomial, it is essential that each term is ordered starting from the highest power to the lowest. For instance:

• The highest power of the variable comes first, such as \(x^2\) before \(x^1\) and before any constant.

So from our solution, \(16x^2 - 25\), we see it is in standard form already:

• It begins with \(16x^2\), representing the highest power,
• and ends with \(-25\), the constant term.

Organizing expressions in this way not only helps in clarity but also makes it easier to perform operations like addition or subtraction against other polynomials.

Polynomial Operations

Polynomial operations encompass addition, subtraction, multiplication, and division of polynomial expressions. Understanding the rules for these operations lays a strong foundation for solving more complex algebraic problems.
For multiplication, which we are given, knowledge of patterns like the difference of squares makes it more efficient.

• When multiplying polynomials, each term in the first polynomial is multiplied by each term in the second.
• However, patterns such as \((a - b)(a + b)\) allow you to bypass detailed term-by-term multiplication.

Another key process is subtraction, which involves changing the sign of every term in the polynomial being subtracted and then adding the results.
Understanding these operations allows you to manipulate and simplify expressions systematically. They form the backbone of algebraic procedures in everything from finding polynomial roots to solving equations.