Problem 41 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using the FOIL method, and simplify. \((3 t-2)(7 t-4)\)

Step-by-Step Solution

Answer

The simplified product is \(21t^2 - 26t + 8\).

Step 1: Understanding FOIL Method

FOIL stands for First, Outer, Inner, Last, which refers to a specific method of multiplying two binomials. The terms of each binomial are paired and multiplied to get corresponding products.

Step 2: Multiply First Terms

Identify the first terms in each binomial, which are \[ 3t \text{ and } 7t. \] Multiply these together:\[ (3t) \cdot (7t) = 21t^2. \]

Step 3: Multiply Outer Terms

The outer terms in the expression are \[ 3t \text{ and } -4. \] Multiply these two terms together:\[ (3t) \cdot (-4) = -12t. \]

Step 4: Multiply Inner Terms

The inner terms in the binomial are \[ -2 \text{ and } 7t. \] Multiply these terms:\[ (-2) \cdot (7t) = -14t. \]

Step 5: Multiply Last Terms

The last terms to multiply are \[ -2 \text{ and } -4. \] Multiply them to get:\[ (-2) \cdot (-4) = 8. \]

Step 6: Combine All Products

Add the results of Steps 2-5 together:\[ 21t^2 - 12t - 14t + 8. \]

Step 7: Simplify the Expression

Combine like terms. The terms \[-12t \text{ and } -14t \] can be added:\[ 21t^2 - 26t + 8. \] This is the simplified expression.

Key Concepts

Understanding Binomials

A binomial is an algebraic expression composed of two terms connected by either a plus or minus sign. Understanding binomials is crucial when learning how to multiply algebraic expressions. In our example, the binomials are \((3t - 2)\) and \((7t - 4)\). These expressions are known as binomials because they each have two distinct terms.

• The term 'bi' indicates two, highlighting the number of terms in the expression.
• Each part of a binomial can contain constants and variables with exponents.

Recognizing binomials helps identify the appropriate mathematical techniques, like the FOIL method, to apply during multiplication. By realizing that you are dealing with binomials, you can methodically break the problem down into manageable parts using strategies like FOIL.

Algebraic Multiplication

Algebraic multiplication entails multiplying terms in algebraic expressions, such as binomials, to simplify them or solve equations. When we multiply two binomials, we can use the FOIL method to simplify this task.
During the FOIL process:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of the binomials.

This technique breaks down the multiplication into simpler steps, ensuring all terms are considered. For example, in \((3t-2)(7t-4)\), by applying FOIL, you execute multiple smaller multiplications, such as multiplying \(3t\) with \(-4\) and \(-2\) with \(7t\). This structured approach makes complex algebraic tasks more manageable, especially for beginners.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that allows us to rewrite expressions in a more concise form. Once you have multiplied the terms within the binomials using the FOIL method, the next step is to simplify the resulting expression by combining like terms.
Combining like terms involves adding or subtracting terms that have the same variable and exponent. In the expression obtained from multiplying \((3t-2)(7t-4)\), we end up with \(21t^2 - 12t - 14t + 8\).

• Identify terms with the same variable and exponent, for instance, \(-12t\) and \(-14t\).
• Add or subtract these terms to simplify the expression further to \(21t^2 - 26t + 8\).

Simplifying not only makes equations easier to read but also prepares expressions for further operations if needed. This refinement of multiplication results is crucial in solving algebraic equations and understanding mathematical relationships.