Problem 16 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(y-3)(y+9)$$

Step-by-Step Solution

Answer

The product is \( y^2 + 6y - 27 \).

Step 1: Recognize the Binomial Product

The expression given is the product of two binomials: \((y - 3)(y + 9)\). We need to multiply these binomials using the distributive property.

Step 2: Apply the Distributive Property

To multiply binomials, apply the distributive property, sometimes referred to as the FOIL method in this context. This means we multiply each term in the first binomial by each term in the second binomial. Expand it as follows:\[ y(y + 9) + (-3)(y + 9) \]

Step 3: Expand Each Term

Now, multiply each term separately:1. \(y \times y = y^2\)2. \(y \times 9 = 9y\)3. \((-3) \times y = -3y\)4. \((-3) \times 9 = -27\)Putting it all together, you get:\[ y^2 + 9y - 3y - 27 \]

Step 4: Combine Like Terms

Combine the like terms, which are the middle terms in the expression:\(9y - 3y = 6y\)The expression simplifies to:\[ y^2 + 6y - 27 \]

Key Concepts

Distributive Property in Binomial Multiplication

When multiplying binomials, the distributive property plays a fundamental role. Let's break down how it works in this context. The distributive property is a form of multiplication that involves distributing each term in the first binomial across each term in the second binomial. In the example
\((y - 3)(y + 9)\), we take each term in \((y - 3)\) and multiply them by each term in \((y + 9)\). This process transforms our initial expression into a series of multiplication tasks:

• First, \(y\) is distributed to both \(y\) and \(9\).
• Second, \(-3\) is also distributed to both \(y\) and \(9\).

These steps ensure every term in the first binomial interacts with every term in the second, creating our expanded expression.

Unfolding Binomial Products with the FOIL Method

The FOIL method is a handy shortcut for using the distributive property on binomials, and it stands for First, Outer, Inner, Last. Each letter represents a pair of terms from the binomials:

• First: Multiply the first terms in each binomial, resulting in \(y \times y = y^2\).
• Outer: Multiply the outer terms, here \(y\) and \(9\), giving \(y \times 9 = 9y\).
• Inner: Multiply the inner terms, which are \(-3\) and \(y\), resulting in \(-3 \times y = -3y\).
• Last: Multiply the last terms of each binomial, \(-3\) and \(9\), giving you \(-3 \times 9 = -27\).

The FOIL method effectively covers all interactions between the terms for a comprehensive expansion of the binomials.

Putting it Together: Combining Like Terms

After expanding the binomials, you'll notice some terms are similar, known as 'like terms.' These need to be combined to simplify the expression. In the expanded expression
\(y^2 + 9y - 3y - 27\), notice the terms with the variable 'y':

• \(9y\) and \(-3y\) are like terms.

Combining these gives \(9y - 3y = 6y\). The simplified expression becomes:

• \(y^2 + 6y - 27\)

Combining like terms is crucial for cleanly finishing binomial multiplication and providing an easy-to-manage result.