Problem 50 - Beginning and Intermediate Algebra

Question

Use FOIL to multiply. $$(6 t+1)(t+7)$$

Step-by-Step Solution

Answer

After applying the FOIL method, the final result is \((6t + 1)(t + 7) = 6t^2 + 43t + 7\).

Step 1: Multiply the First terms

We begin by multiplying the first terms of each binomial, which are 6t and t. \(6t \cdot t = 6t^2\)

Step 2: Multiply the Outer terms

Now, we multiply the outer terms of each binomial, which are 6t and 7. \(6t \cdot 7 = 42t\)

Step 3: Multiply the Inner terms

We proceed to multiply the inner terms of each binomial, which are 1 and t. \(1 \cdot t = t\)

Step 4: Multiply the Last terms

Lastly, we multiply the last terms of each binomial, which are 1 and 7. \(1 \cdot 7 = 7\)

Step 5: Combine the results

Now that we have the products from each step, we combine them to form the final expression: \(6t^2 + 42t + t + 7\)

Step 6: Simplify the expression

Simplify the expression by combining like terms: \(6t^2 + 43t + 7\) Thus, the final result after applying FOIL to the given expression is: \((6t + 1)(t + 7) = 6t^2 + 43t + 7\)

Key Concepts

Polynomial Multiplication

Polynomial multiplication is a fundamental concept in algebra that involves expanding expressions to form a new polynomial. In our case, we use the FOIL method to multiply two binomials, like \((6t + 1)(t + 7)\). FOIL is an acronym that guides us through multiplying the First, Outer, Inner, and Last terms of the binomials.

• First: Multiply the first terms of each binomial (\(6t \cdot t = 6t^2\)).
• Outer: Multiply the outer terms (\(6t \cdot 7 = 42t\)).
• Inner: Multiply the inner terms (\(1 \cdot t = t\)).
• Last: Multiply the last terms (\(1 \cdot 7 = 7\)).

This structured method helps in organizing the operation and ensuring that no term is forgotten. By following these steps, you end up with the expanded polynomial expression ready for the next step—simplification.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations such as addition, subtraction, multiplication, and division. In the exercise with an expression like \((6t + 1)(t + 7)\), the FOIL method helps expand this combined expression into a form that makes each part of it easier to handle.

Here, we're dealing with binomials, which are polynomials that contain exactly two terms. Binomials are a simplified version of algebraic expressions that we often encounter in problems involving polynomial multiplication.

Algebraic expressions allow us to succinctly encode mathematical relationships and operations. Knowing how to manipulate and simplify them is critical in problem-solving because it lays the groundwork for understanding complex mathematical concepts.

Like Terms Simplification

Once you have used the FOIL method and expanded the polynomial, it’s essential to simplify the expression by combining like terms.

Like terms are terms within a polynomial that have the same variable raised to the same power, such as \(42t\) and \(t\) in the FOIL example.

To simplify, add the coefficients of the like terms, producing a cleaner and more concise expression:

• Combine \(42t\) and \(t\) to get \(43t\)

The final simplified expression from our exercise is \(6t^2 + 43t + 7\).
Understanding and mastering the process of combining like terms is essential for evaluating and reducing expressions accurately, making it a critical skill in algebra.