Problem 23 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(21t^2 - 26t + 8\)

Step 1: Apply FOIL Method - Multiply First Terms

Start by multiplying the first terms in each binomial: \( 3t \times 7t = 21t^2 \)

Step 2: Apply FOIL Method - Multiply Outer Terms

Next, multiply the outer terms of the binomials: \( 3t \times (-4) = -12t \)

Step 3: Apply FOIL Method - Multiply Inner Terms

Then, multiply the inner terms: \((-2) \times 7t = -14t \)

Step 4: Apply FOIL Method - Multiply Last Terms

Finally, multiply the last terms in each binomial: \((-2) \times (-4) = 8 \)

Step 5: Add and Simplify

Combine the results from each step: \( 21t^2 + (-12t) + (-14t) + 8 \).Combine like terms: \( 21t^2 - 26t + 8 \).

Key Concepts

Algebraic Expressions

When we talk about algebraic expressions, we're dealing with a combination of numbers, variables, and operations. Variables are symbols, often letters, that stand in for unknown or changeable values. For example, in the expression \(3t - 2\), "3" is a coefficient, "t" is a variable, and "-2" is a constant. Algebraic expressions represent various relationships and can be manipulated to solve problems or to simplify calculations. They play a crucial role in mathematics and are foundational for understanding more complex math topics.

Polynomial Multiplication

Polynomial multiplication involves multiplying expressions that have multiple terms, such as the two binomials in our exercise: \((3t - 2)(7t - 4)\). One of the key methods for multiplying these is the FOIL method, which stands for First, Outer, Inner, Last, referring to the pairs of terms you multiply together. - **First Terms**: Multiply the first term in each binomial. In our example, this is \(3t \times 7t = 21t^2\).- **Outer Terms**: Multiply the first term of the first binomial by the last term of the second binomial: \(3t \times (-4) = -12t\).- **Inner Terms**: Multiply the last term of the first binomial by the first term of the second binomial: \((-2) \times 7t = -14t\).- **Last Terms**: Multiply the last terms in each binomial: \((-2) \times (-4) = 8\).By using the FOIL method, we ensure that every part of each binomial is multiplied by every part of the other binomial, allowing us to keep track of all the pieces and simplifying the entire process.

Simplification of Expressions

Simplifying mathematical expressions involves reducing them to their simplest form. After applying the FOIL method to multiply the expressions \((3t - 2)(7t - 4)\), we end up with several terms. These terms are \(21t^2\), \(-12t\), \(-14t\), and \(8\).To simplify: - Look for like terms, which are terms that are similar, such as \(-12t\) and \(-14t\). These can be combined because they have the same variable part "t."- Perform the addition, which in our case, results in \(-26t\) when combining \(-12t + -14t\).- Combine all simplified terms to get the final expression: \(21t^2 - 26t + 8\).Simplifying expressions makes them more manageable and allows for easier interpretation and use in further mathematical procedures or real-world applications.