Problem 18 - 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. $$(n+3)(n-12)$$

Step-by-Step Solution

Answer

The product is \(n^2 - 9n - 36\).

Step 1: Identify the expression type

The given expression to find the product is \((n+3)(n-12)\). This is a multiplication of two binomials. We will use the distributive property, also known as the FOIL method (First, Outside, Inside, Last), to find the product.

Step 2: Apply the FOIL Method

Multiply the terms using FOIL: - **First**: Multiply the first terms in each binomial: \(n \times n = n^2\).- **Outside**: Multiply the outside terms: \(n \times (-12) = -12n\).- **Inside**: Multiply the inside terms: \(3 \times n = 3n\).- **Last**: Multiply the last terms in each binomial: \(3 \times (-12) = -36\).

Step 3: Combine Like Terms

Combine the results from the FOIL method: - Add the like terms \(-12n\) and \(3n\): \[n^2 - 12n + 3n - 36\] Simplify the combined terms: \[n^2 - 9n - 36\]

Step 4: Write the Final Product

The product of \((n+3)(n-12)\) is the simplified version obtained in the previous step: \[n^2 - 9n - 36\].

Key Concepts

FOIL Method

The FOIL method is a straightforward approach to multiplying two binomials. It stands for First, Outside, Inside, and Last, which refers to the order in which the terms in each binomial are multiplied. This technique is useful as it ensures you account for all the necessary products to obtain the correct result. Let's break it down further:

• First: Multiply the first term in each binomial, which in our example is \(n \times n = n^2\).
• Outside: Multiply the outermost terms, giving us \(n \times (-12) = -12n\).
• Inside: Multiply the innermost terms, resulting in \(3 \times n = 3n\).
• Last: Multiply the last term in each binomial, leading to \(3 \times (-12) = -36\).

By applying the FOIL method, you can confidently multiply any two binomials and simplify to find the final expression.

Distributive Property

The distributive property is a fundamental algebraic principle that allows you to multiply a single term by each term within a parenthesis. It's expressed as \(a(b + c) = ab + ac\). This property is the foundation of the FOIL method and is essential in algebra.
For example, when solving \((n+3)(n-12)\), think of the expression as:

• Multiply \(n\) by each term inside the second binomial: yielding \(n^2 - 12n\).
• Multiply \(3\) by each term inside the second binomial: resulting in \(3n - 36\).

After that, these results are added together. Understanding the distributive property will greatly enhance your ability to work with polynomials.

Combining Like Terms

Once you apply the FOIL method or distributive property, your next step is to combine like terms. This means you should add or subtract terms that have the exact same variable and exponent.
In our case, after applying FOIL to \((n+3)(n-12)\), we have the terms \(n^2\), \(-12n\), \(3n\), and \(-36\). Here, \(-12n\) and \(3n\) are like terms because they both include the variable \(n\) raised to the first power.

• Combining \(-12n\) and \(3n\) yields \(-9n\).

This simplification step reduces the expression to \(n^2 - 9n - 36\). Properly combining like terms is crucial to achieving the most concise expression possible.

Special Patterns in Algebra

Algebra includes several special patterns that can simplify the process of multiplying certain types of binomials. Recognizing these patterns can save you time and effort. Here are a few common patterns:

• Square of a Binomial: \((a + b)^2 = a^2 + 2ab + b^2\)
• Difference of Squares: \((a + b)(a - b) = a^2 - b^2\)
• Perfect Square: When you recognize that \(a^2 \pm 2ab + b^2\) resembles a binomial squared, it can be factored accordingly.

While \((n+3)(n-12)\) isn't a special pattern, knowing these patterns prepares you for quicker factoring and expanding elsewhere in algebra. Always look for these clues to streamline your work.