Problem 14 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \((5 b+2)(5 b-2)\)

Step-by-Step Solution

Answer

\((5b+2)(5b-2) = 25b^2 - 4\) in simplest form.

Step 1: Understanding the Formula

The expression \((a + b)(a - b)\) represents a difference of squares. The formula for simplifying a difference of squares is: \[a^2 - b^2\] This means we need to recognize each "a" and "b" in our expression \((5b+2)(5b-2)\). Here, we identify: - "a" as \(5b\)- "b" as \(2\).

Step 2: Applying Difference of Squares

Now, apply the difference of squares formula \[(a + b)(a - b) = a^2 - b^2\]We have:- \(a = 5b\)- \(b = 2\)Substitute these into the formula: \[(5b+2)(5b-2) = (5b)^2 - 2^2\]

Step 3: Calculating the Squares

Calculate each square:- \((5b)^2 = 25b^2\)- \(2^2 = 4\)Now, substitute these values back into the formula: \[25b^2 - 4\]

Step 4: Final Simplification

The expression \[25b^2 - 4\] is already in its simplest form, as both terms are simplified and there are no like terms to combine. This is the final result of the operation.

Key Concepts

Polynomial Multiplication

In algebra, one of the core operations is polynomial multiplication. This involves multiplying expressions with multiple terms, referred to as polynomials. When you encounter a problem like

• a multiplication of expressions
• such as
• \((5b+2)(5b-2)\),

it’s essentially asking you to apply this process.
The special case here is the "difference of squares" which provides a shortcut for multiplying binomials in the form of

• \((a+b)(a-b)\).

This pattern allows you to quickly simplify the expression by using the formula

• \(a^2 - b^2\).

Thus, polynomial multiplication is not just about multiplying individual terms, but also recognizing patterns that simplify the process.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra, as it involves reducing expressions into their simplest form. This process can make a complex problem much easier to handle and interpret. For instance, when dealing with

• the difference of squares in this problem,
• you use the calculated squares
• \((5b)^2 = 25b^2\) and
• \(2^2 = 4\),

to substitute back into the expression as

• \(25b^2 - 4\).

This final expression is simpler since each term is completely reduced, and there are no more operations left to perform between them.
The essence of simplifying expressions is

• identifying patterns and
• grouping like terms

which streamlines the expression into a form that is more manageable.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra, consisting of variables, numbers, and arithmetic operations. In the exercise, we see the expression

• \((5b+2)(5b-2)\),

which combines these elements into a structured format that can be manipulated using algebraic rules. An important kind of expression involves polynomials, which are sums of terms with variables raised to whole-number exponents.
This particular problem of multiplying binomials highlights how different kinds of polynomial expressions can be simplified using algebraic strategies such as the formula for a difference of squares.
Understanding algebraic expressions involves recognizing

• the role of each component,
• the impact of operations between them,

and employing known formulas to simplify or solve the expression accurately. This forms a cornerstone of numerical problem-solving in algebra.