Problem 17 - 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. $$(n+2)(n-7)$$

Step-by-Step Solution

Answer

The product is \(n^2 - 5n - 14\).

Step 1: Set Up the Problem

We are tasked with multiplying the binomials \((n+2)(n-7)\). We will use the distributive property (also known as the FOIL method for binomials) to find the product.

Step 2: Apply the FOIL Method

The FOIL method stands for First, Outer, Inner, Last, which helps us remember how to distribute each term:- **First**: Multiply the first terms: \(n \times n = n^2\).- **Outer**: Multiply the outer terms: \(n \times -7 = -7n\).- **Inner**: Multiply the inner terms: \(2 \times n = 2n\).- **Last**: Multiply the last terms: \(2 \times -7 = -14\).

Step 3: Combine Like Terms

Now, we combine the terms obtained from the FOIL method:\(n^2 - 7n + 2n - 14\).Combine the \(-7n\) and \(+2n\):\(n^2 - 5n - 14\).

Step 4: Write the Final Product

The expression \(n^2 - 5n - 14\) is the product of the binomials \((n+2)(n-7)\).

Key Concepts

FOIL Method

Multiplying binomials can seem tricky, but the FOIL method makes it simple. The acronym FOIL stands for First, Outer, Inner, Last.
It is a handy way to remember which terms to multiply.
Let's break it down:

• First: Multiply the first terms from each binomial. In this case, the first terms are both "n," so we have: \(n \times n = n^2\).
• Outer: Multiply the outer terms. Here, the terms are "n" and "-7," giving: \(n \times -7 = -7n\).
• Inner: Multiply the inner terms. The inner terms "2" and "n" must be multiplied, resulting in: \(2 \times n = 2n\).
• Last: Multiply the last terms in each binomial. For our example, "2" and "-7" multiply to: \(2 \times -7 = -14\).

Using the FOIL method is like peeling layers off an onion, one process at a time, ensuring no multiplication is missed.

Distributive Property

The FOIL method is a specific type of the distributive property. The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number, and then adding those products together.
For binomials \((a + b)(c + d)\), we use:

• First, we distribute "a": \((a \times c) + (a \times d)\).
• Then, we distribute "b": \((b \times c) + (b \times d)\).

After writing out these products, we get: \((a \times c) + (a \times d) + (b \times c) + (b \times d)\). This is a fundamental concept in algebra, underpinning much of polynomial arithmetic. Each term from the first binomial is distributed through each term of the second, laying the foundation for polynomial multiplication.

Polynomial Multiplication

Polynomial multiplication involves combining terms from two or more polynomials to produce a single polynomial. When dealing with binomials like \((n+2)(n-7)\), you essentially apply the distributive property twice, covering every term.
Let's summarize polynomial multiplication through this example:

• The polynomial starts as two binomials: \((n + 2)\) and \((n - 7)\).
• We use the FOIL method (a specialized case) to multiply each term.
• Combine like terms: after using the FOIL method, we find similarity in \(-7n + 2n\) to simplify to \(-5n\).

Thus, the product of \((n+2)(n-7)\) ends up as a quadratic polynomial \(n^2 - 5n - 14\). Mastery of polynomial multiplication is crucial in higher algebra and calculus, transforming complex expressions into manageable forms.