Problem 83 - Algebra A Combined Function

Question

Multiply. $$ (b+1)(b-4) $$

Step-by-Step Solution

Answer

The result of multiplying \((b+1)(b-4)\) is \(b^2 - 3b - 4\).

Step 1: Understand the Problem

We're asked to multiply two binomials: \((b + 1)(b - 4)\). This involves using the distributive property, also known as the FOIL (First, Outer, Inner, Last) method, to expand the expression.

Step 2: Apply the FOIL Method

Apply the FOIL method to expand: First, multiply the first terms \((b \cdot b)\), Outer \((b \cdot -4)\), Inner \((1 \cdot b)\), and Last \((1 \cdot -4)\).

Step 3: Calculate Each Part

Calculate each part: First: \(b \cdot b = b^2\), Outer: \(b \cdot -4 = -4b\), Inner: \(1 \cdot b = b\), Last: \(1 \cdot -4 = -4\).

Step 4: Combine Like Terms

Combine the results from Step 3: \(b^2 - 4b + b - 4\). Then, combine the middle terms: \(-4b + b = -3b\).

Step 5: Write the Final Solution

The expanded form of the multiplication \((b + 1)(b - 4)\) is \(b^2 - 3b - 4\).

Key Concepts

FOIL Method

The FOIL method is a handy way to multiply two binomials. Each letter in "FOIL" stands for a pair of terms to multiply: **First**, **Outer**, **Inner**, **Last**. This method helps you remember to multiply each term in one binomial with each term in the other.
Here’s how it breaks down for our example

• **First**: Multiply the first terms of each binomial. Here, that would be \(b \times b\), which equals \(b^2\).
• **Outer**: Multiply the outer terms. That's \(b\) from the first binomial and \(-4\) from the second, giving us \(-4b\).
• **Inner**: Then multiply the inner terms, which are \(1\) from the first and \(b\) from the second binomial. This results in \(b\).
• **Last**: Lastly, multiply the last terms from each binomial, \(1\) and \(-4\), to get \(-4\).

By following these steps, you expand the binomial multiplication into a set of four terms: \(b^2 - 4b + b - 4\). Remember, the FOIL method is just a structured way of using the distributive property, but with a twist that makes it perfect for binomials.

Distributive Property

The distributive property is a fundamental principle in algebra that lets us multiply a single term across terms inside a bracket. When dealing with binomials, it allows us to distribute each term of one binomial to every term of the other. It often seems complex, but here’s why it’s a crucial tool in binomial multiplication.
The distributive property states that \(a(b+c) = ab + ac\). This means you multiply \(a\) with both \(b\) and \(c\). For the binomials \((b+1)(b-4)\):

• Think of the first binomial \(b+1\) as \(a\) and distribute it across the second binomial \(b-4\).
• So, you first multiply \((b)\) by each term \((b-4)\), resulting in \(b^2 - 4b\).
• Then multiply \((+1)\) with each term \((b-4)\), ending up with \(b-4\).

Combining these, we achieve \((b+1)(b-4) = b^2 - 4b + b - 4\), beautifully demonstrating the power of the distributive property in simplifying expressions.

Like Terms

Like terms are terms in an algebraic expression that have the same variable raised to the same power. They are essential to identify because they can be combined to simplify expressions. Understanding like terms can help streamline the process of solving and simplifying equations.
In the expression from our binomial multiplication, \(b^2 - 4b + b - 4\), you will notice:

• \(b^2\) is its own term because there is no other \(b^2\) term to combine it with.
• \(-4b\) and \(b\) are like terms because both terms have the variable \(b\) raised to the same power (1).
• When we combine \(-4b\) and \(b\), we simplify them to \(-3b\).
• Finally, \(-4\) is constant and stays as is since there are no other constant terms to combine with it.

Through combining like terms, the expression is further simplified to \(b^2 - 3b - 4\), which is the clean, expanded result of the binomial multiplication. Recognizing and combining like terms is a critical step in solving algebraic equations.