Problem 50 - Algebra A Combined Function

Question

Multiply. $$ (8-7 x)(8+7 x) $$

Step-by-Step Solution

Answer

\((8-7x)(8+7x) = 64 - 49x^2\).

Step 1: Recognize the formula

The expression \((8 - 7x)(8 + 7x)\) is of the form \((a - b)(a + b)\). This is a difference of squares, which can be simplified using the formula \((a - b)(a + b) = a^2 - b^2\).

Step 2: Identify the terms

Identify \(a = 8\) and \(b = 7x\) from the given expression \((8 - 7x)(8 + 7x)\).

Step 3: Apply the difference of squares formula

Substitute the identified terms \(a = 8\) and \(b = 7x\) into the formula \(a^2 - b^2\): \[ a^2 = 8^2 = 64 \] \[ b^2 = (7x)^2 = 49x^2 \] Therefore, \[ (8 - 7x)(8 + 7x) = 64 - 49x^2 \]

Step 4: Write the final expression

The product of the expression using the difference of squares is: \[ 64 - 49x^2 \].

Key Concepts

Algebraic Expressions

Algebraic expressions are a key part of algebra and mathematics as a whole. They involve numbers, variables, and operations that combine to form expressions. In the exercise provided, we deal with a product of two algebraic expressions:

• Expression One: \(8 - 7x\)
• Expression Two: \(8 + 7x\)

In many cases, expressions like these hold special forms that can simplify our calculations. For this problem, the expressions reflect a pattern called the difference of squares. Understanding this form helps us manipulate and simplify expressions quickly. Remember that variables like \(x\) can change, which means our resulting expression represents a whole range of values depending on what \(x\) we choose.

Polynomial Multiplication

Polynomial multiplication involves combining each term of one polynomial with each term of the other. When we multiply expressions like \((8 - 7x)(8 + 7x)\), we generally distribute each term in the first polynomial across the terms in the second one. However, there's an easier approach if the expressions fit a known pattern, like the difference of squares. In this exercise:

• Count each term in both binomials; both have two terms.
• Normally, every term in the first binomial would multiply with every term in the second, a process following the FOIL method (First, Outer, Inner, Last).
• This method applies each multiplication individually, resulting in four terms, later simplified.

However, noticing the specific structure in the problem lets us use a formula to simplify this quickly, highlighting the importance of recognizing algebraic patterns.

Mathematical Formulas

Mathematical formulas are tools that allow us to solve intricate problems without doing all the calculations manually. The formula used in our scenario is the difference of squares: \[(a - b)(a + b) = a^2 - b^2\]Applied here, this formula identifies - \(a = 8\) and - \(b = 7x\).We calculate each square individually:

• \(a^2 = 8^2 = 64\)
• \(b^2 = (7x)^2 = 49x^2\)

Substituting back, - the entire expression simplifies to \(64 - 49x^2\). This simplification shows how formulas condense multiple steps into a single process, making arithmetic more efficient and showcasing the elegance within algebraic structures.