Problem 74 - Algebra 2

Question

Find each product. $$ (a+2)(a-9) $$

Step-by-Step Solution

Answer

The product is \(a^2 - 7a - 18\).

Step 1: Identify the Structure

Recognize that this is a multiplication of two binomials, each of the form \((a + b)(a + c)\). Our specific binomials are \((a + 2)\) and \((a - 9)\).

Step 2: Apply the Distributive Property

Use the distributive property, also known as the FOIL method (First, Outer, Inner, Last), to expand the product \((a + 2)(a - 9)\).

Step 3: First Terms Multiplication

Multiply the first terms of each binomial: \(a \times a = a^2\).

Step 4: Outer Terms Multiplication

Multiply the outer terms: \(a \times (-9) = -9a\).

Step 5: Inner Terms Multiplication

Multiply the inner terms: \(2 \times a = 2a\).

Step 6: Last Terms Multiplication

Multiply the last terms: \(2 \times (-9) = -18\).

Step 7: Combine the Results

Add all the results from the previous steps together: \(a^2 - 9a + 2a - 18\).

Step 8: Simplify the Expression

Combine like terms in the expression: combine \(-9a\) and \(2a\) to get \(-7a\). This gives a final expression of \(a^2 - 7a - 18\).

Key Concepts

FOIL Method

The FOIL Method is a handy trick for multiplying two binomials. It stands for First, Outer, Inner, and Last. These terms refer to the positions of the terms in the binomials that you will be multiplying together.

• **First:** Multiply the first term in each binomial. In our example, that's \(a \times a\), giving us \(a^2\).
• **Outer:** Next, multiply the outer terms. Here, these are \(a\) from \((a + 2)\) and \(-9\) from \((a - 9)\), resulting in \(-9a\).
• **Inner:** Multiply the inner terms: \(2 \times a\), which equals \(2a\).
• **Last:** Finally, multiply the last terms in each binomial: \(2 \times -9\) equals \(-18\).

Using FOIL helps to ensure that you've accounted for every part of the multiplication process. It's a structured approach that breaks down the multiplication into manageable steps. Once you've calculated the results from these four operations, you simply add them together to form an expanded polynomial expression.

Distributive Property

The distributive property is a valuable mathematical rule that simplifies the process of multiplying a single term by a binomial or multiplying binomials. In essence, it states that \(a(b + c) = ab + ac\).
When applied to binomials like \((a + 2)(a - 9)\), the distributive property helps to see how each term in the first binomial will multiply with each term in the second.
Here's how it works:

• First, take the first term in the first binomial \(a\) and distribute it across \((a - 9)\), resulting in \(a^2 - 9a\).
• Next, apply the second term of the first binomial \(2\) to \((a - 9)\) as well, giving us \(2a - 18\).

Once distributed, you combine like terms to simplify the expression further. This method is identical to the structure of the FOIL method but expands the understanding by emphasizing the essence of distribution over simple position-based multiplications.

Simplifying Expressions

Simplifying expressions is the final stage in multiplying binomials. After using either the FOIL Method or the distributive property to expand the expression, you'll often have terms that can be combined.
In our example, the expanded form \(a^2 - 9a + 2a - 18\) can be simplified by combining like terms.

• **Combine Like Terms:** Here, we have two 'a' terms: \(-9a\) and \(2a\). When these are combined, they simplify to \(-7a\).
• **Final Simplified Expression:** After solving, the expression becomes \(a^2 - 7a - 18\).

Simplifying expressions is essential because it makes them cleaner and easier to interpret. In mathematics, a simplified expression is usually preferred, as it offers the most straightforward representation of the equation.