Problem 34 - College Algebra

Question

For the following exercises, multiply the binomials. $$(25 b+2)(25 b-2)$$

Step-by-Step Solution

Answer

The product is \(625b^2 - 4\).

Step 1: Recognize the Pattern

The given expression, \((25b + 2)(25b - 2)\), fits the form of a difference of squares, \((a + b)(a - b) = a^2 - b^2\). This pattern simplifies multiplication significantly.

Step 2: Identify Values for the Formula

Recognize that in the expression \((25b + 2)(25b - 2)\), \(a = 25b\) and \(b = 2\). We will use these values in the formula for difference of squares.

Step 3: Apply the Difference of Squares Formula

Substitute the identified values into the difference of squares formula: \((a)^2 - (b)^2 = (25b)^2 - (2)^2\).

Step 4: Calculate the Squares

First, calculate \((25b)^2 = 625b^2\), and then calculate \((2)^2 = 4\).

Step 5: Simplify the Expression

Substitute the squared values back into the expression: \(625b^2 - 4\). This is the simplified product of the binomials.

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations. It's like a phrase in mathematics that can contain a mix of addition, subtraction, multiplication, and division with variables that stand for unknown values.

In the expression \(25b + 2\), \('b'\) is the variable, and \(25\) and \(2\) are coefficients and constants, respectively. The coefficients multiply the variable, and constants are just standalone numbers.

Expressions can be simplified or manipulated to make calculations easier. They can take many forms, but they all follow the rules of arithmetic. Recognizing different forms and patterns in algebraic expressions, such as the difference of squares, is essential for solving problems quickly and efficiently.

Exploring Binomial Multiplication

Binomial multiplication involves multiplying two binomials, which are algebraic expressions containing two terms. The binomials in our example are \(25b + 2\) and \(25b - 2\).

One might initially use the FOIL method to expand binomials, multiplying each term in the first binomial by each term in the second. However, the term here can be simplified using the difference of squares method. Here's why: the expressions \(a + b\) and \(a - b\) multiply to form \[(a + b)(a - b) = a^2 - b^2\].

This shortcut occurs because the inner products cancel each other out, leaving only a difference of squares, making calculations faster and less error-prone. Knowing when to use this form can save time and effort in solving algebraic problems.

Polishing Polynomial Simplification

Simplifying polynomials is a crucial step in solving algebraic equations. The final simplified form makes the expressions more interpretable and easier to handle. In our problem, we simplify the result of the multiplication of two binomials.

Using the difference of squares, the expression \( (25b + 2)(25b - 2) \) becomes \( (25b)^2 - (2)^2 \).

• First, calculate \((25b)^2\). This equals \((25)^2(b)^2 = 625b^2\).
• Next, calculate \(2^2\), which equals \(4\).

The end result is \(625b^2 - 4\), a polynomial in its simplest form. Here, nothing further can be combined or factored, indicating completion. Mastery of polynomial simplification helps uncover solutions quickly, tackling equations with efficiency.