Problem 103 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (5+2 b)(5-2 b) $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression (5+2b)(5-2b). Answer: The simplified expression is 25 - 4b^2.

Step 1: Identify the terms

First, let's identify the terms in the given expression: $$ (5+2b)(5-2b) $$ Here, we have two binomials, \((5+2b)\) and \((5-2b)\).

Step 2: Apply the distributive property (FOIL method)

Now, we will multiply each term of the first binomial with each term of the second binomial using the FOIL method (First, Outer, Inner, Last): First: \(5 \cdot 5\) Outer: \(5 \cdot (-2b)\) Inner: \(2b \cdot 5\) Last: \(2b \cdot (-2b)\)

Step 3: Perform the multiplication

Perform the multiplication for each term found in Step 2: First: \(5 \cdot 5 = 25\) Outer: \(5 \cdot (-2b) = -10b\) Inner: \(2b \cdot 5 = 10b\) Last: \(2b \cdot (-2b) = -4b^2\)

Step 4: Combine like terms

Now, we will combine the like terms resulting from the multiplication: $$25 - 10b + 10b - 4b^2$$ The terms \(-10b\) and \(+10b\) cancel each other out as they add up to zero. So, we are left with: $$25 - 4b^2$$

Step 5: Write the final simplified expression

The simplified expression for the given problem is: $$ (5+2b)(5-2b) = 25 - 4b^2 $$

Key Concepts

FOIL Method

The FOIL method stands for First, Outer, Inner, Last, and it's a technique used to simplify the multiplication of two binomials. Imagine you want to multiply \( (a+b)(c+d) \). The FOIL method guides you to multiply the terms in a specific order:

• First: Multiply the first terms of each binomial (\(a \times c\)).
• Outer: Multiply the outer terms (\(a \times d\)).
• Inner: Multiply the inner terms (\(b \times c\)).
• Last: Multiply the last terms of each binomial (\(b \times d\)).

After you have these products, you add them together to get the expanded form before simplifying further by combining like terms.

Distributive Property

The distributive property is a foundational algebraic principle that allows us to multiply a single term across the terms within a parenthesis. It is used both independently and as part of the FOIL method. For example, if we have \( a(b + c) \), the distributive property would expand this to \( ab + ac \).

In the context of the exercise \( (5+2b)(5-2b) \), we applied the distributive property through the FOIL method. Each term in the first binomial is distributed, or multiplied, across each term in the second binomial, resulting in four terms that are then combined into the final simplified expression.

Combine Like Terms

Once you have used the FOIL method to multiply the binomials and expanded the expression into several terms, it's time to combine like terms. Like terms are terms that have the same variables to the same power. For example, \(3b\) and \(5b\) are like terms, but \(3b\) and \(3b^2\) are not.

In our exercise, after applying FOIL, we had the terms \(25\), \( -10b\), \(10b\), and \( -4b^2\). Notice \( -10b\) and \(10b\) are like terms, and they cancel each other out as they sum to zero. We're then left with \(25 - 4b^2\), which is the simplified form of the original expression.

Binomials Multiplication

Multiplying binomials, like \( (x + y)(x - y) \), is a process where each term of the first binomial is multiplied by each term of the second. The FOIL method is a specific case of binomial multiplication, designed to systematize the process and ensure you don't miss any terms. After using FOIL, binomial multiplication often results in a quadratic expression, which you might recognize by its standard form \( ax^2 + bx + c \).

In our example, we've multiplied the binomials \( (5+2b)(5-2b) \) resulting in \( 25 - 4b^2 \), which is now a simpler quadratic expression, showcasing how the product of a sum and difference of the same terms always results in the difference of two squares.