Problem 42 - 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. $$(4 t+6)^{2}$$

Step-by-Step Solution

Answer

The product is \(16t^2 + 48t + 36\).

Step 1: Identify the Expression Type

The expression \((4t + 6)^2\) is in the form \((a + b)^2\), which is a perfect square binomial.

Step 2: Recall Perfect Square Formula

Use the perfect square formula: \((a + b)^2 = a^2 + 2ab + b^2\). Identify \(a = 4t\) and \(b = 6\).

Step 3: Calculate Each Part of the Formula

Compute each term separately: \(a^2 = (4t)^2 = 16t^2\), \(2ab = 2(4t)(6) = 48t\), and \(b^2 = 6^2 = 36\).

Step 4: Combine the Terms

Combine all these parts to get the expanded form of the expression: \(16t^2 + 48t + 36\).

Key Concepts

Multiplying Binomials: A Simple Approach

When it comes to multiplying binomials, you can think of it as multiplying two expressions, often containing variables and constants. Consider a binomial like \((a + b)\) - it is simply a two-term expression where \(a\) and \(b\) are parts separated by a plus or minus sign. When you multiply binomials, you're taking two of these expressions and finding their product.
But rather than multiplying each term directly, a neat trick is to use "special patterns" that make calculation easier. For example, the "perfect square" pattern is one such shortcut. This pattern states that when you have an expression of the type \((a + b)^2\), rather than multiplying it naïvely term by term, you can use a formula to find the result.
Understanding this can save you time and reduce errors when dealing with binomials.

• A binomial is an algebraic expression with two terms.
• Patterns like the perfect square simplify multiplication.

Expanding Expressions and Their Patterns

Expanding expressions involves transforming them from a compact, usually squared, form to a longer, linear form. In the context of the perfect square binomial, we expand by using the identity \((a + b)^2 = a^2 + 2ab + b^2\).
This simple formula helps break a complex multiplication into smaller, manageable parts, separating each term involved.
When expanding, follow these steps:

• Identify \(a\) and \(b\) in \((a + b)^2\).
• Calculate \(a^2\), \(2ab\), and \(b^2\) individually.
• Combine the results to form a linear expression.

For instance, in the expression \((4t + 6)^2\), identify \(a = 4t\) and \(b = 6\). Applying the formula: \(a^2 = 16t^2\), \(2ab = 48t\), and \(b^2 = 36\). Combine these results to get the expanded form: \(16t^2 + 48t + 36\). This method ensures you correctly account for all terms.

Using Formulas in Algebra: The Perfect Square Example

In algebra, formulas are powerful tools that simplify complex problems. Specifically, using the perfect square formula makes expands certain types of binomials much easier. The formula \((a + b)^2 = a^2 + 2ab + b^2\) is crucial in transforming perfect square binomials to their expanded form.
These kinds of formulas serve as blueprints. They allow you to quickly break down expressions without fully reworking them from scratch. Applying such formulas requires identifying components correctly - recognizing \(a\) and \(b\), computing their square, and their product. Once you grasp these steps, multiplying and expanding binomials becomes straightforward.
In our example, \((4t + 6)^2\), the formula shows its value by letting us calculate the expanded form efficiently. It demonstrates:

• The importance of breaking down complex tasks into steps.
• Why memorizing key formulas can save both time and effort.
• How algebraic rules help achieve precise and fast computation results.

When you know which formula to apply, algebra becomes less daunting and more predictable.