Problem 23 - 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. $$(x-6)(x-8)$$

Step-by-Step Solution

Answer

The product is \(x^2 - 14x + 48\).

Step 1: Identify the Binomial Expression

The given expression is \((x-6)(x-8)\). This is a product of two binomials, each in the form \((x-a)(x-b)\), where \(a = 6\) and \(b = 8\).

Step 2: Apply the FOIL Method

The FOIL method stands for First, Outer, Inner, Last, and is used to multiply binomials. Apply each part as follows:- **First:** Multiply the first terms: \(x \times x = x^2\)- **Outer:** Multiply the outer terms: \(x \times (-8) = -8x\)- **Inner:** Multiply the inner terms: \((-6) \times x = -6x\)- **Last:** Multiply the last terms: \((-6) \times (-8) = 48\)

Step 3: Combine Like Terms

To get the final expression, combine like terms from the results:Combine the \(-8x\) and \(-6x\) to get \(-14x\).Thus, the expression becomes:\[x^2 - 14x + 48\]

Step 4: Write the Final Simplified Expression

The product of \((x-6)(x-8)\) using the steps above results in:\[x^2 - 14x + 48\]

Key Concepts

FOIL Method

The FOIL Method is a simple and effective technique to multiply two binomials.
When binomials, such as \((x-6)\) and \((x-8)\), need to be multiplied, we apply the FOIL strategy which involves multiplying the First, Outer, Inner, and Last terms. This method ensures that every term in the first binomial is multiplied by every term in the second binomial. To break it down:

• **First**: Multiply the first terms of each binomial: \(x imes x = x^2\).
• **Outer**: Multiply the outer terms: \(x imes (-8) = -8x\).
• **Inner**: Multiply the inner terms: \((-6) imes x = -6x\).
• **Last**: Multiply the last terms of each binomial: \((-6) imes (-8) = 48\).

Each of these four products represents an essential part of expanding the binomial expression. The arrangement of FOIL helps ensure that learners remember to account for all possible products, giving a complete multiplication of the binomials.

Binomial Products

Binomial products occur when two binomial expressions are multiplied together, resulting in a trinomial or sometimes other types of polynomials.
In our example, the binomials \((x-6)\) and \((x-8)\) produce a trinomial when multiplied using the FOIL method.This process, at its core, involves expanding the expression. Expansion means writing the multiplication in an extended form and eventually simplifying it.
For \((x-6)(x-8)\), each step in the FOIL method gives us products that we add together:

• The product of the first terms is added to the products of the outer, inner, and last terms.
• After fully applying FOIL, the intermediate expression becomes \(x^2 - 8x - 6x + 48\).

This is often the moment when students see the binomials transform into a more expanded and visible polynomial form, bridging to the next step of simplifying.

Combining Like Terms

Once the binomial multiplication is expanded, the next step is to combine like terms. Like terms, in algebra, are terms that have identical variable components raised to the same power.
In our trinomial \(x^2 - 8x - 6x + 48\), we identify like terms:

• Terms with \(x\) as a variable: Here, \(-8x\) and \(-6x\) are like terms and can be combined because they share the same variable and exponent.
• The \(x^2\) term and the constant \(48\) do not have like terms to combine with in this example.

Adding \(-8x\) and \(-6x\) results in \(-14x\), giving the simplified expression \(x^2 - 14x + 48\). Combining like terms is crucial for simplifying expressions to their simplest form, allowing us to see the polynomial's true structure clearly.