Problem 48 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using the FOIL method, and simplify. \((6 u+5 v)(u-2 v)\)

Step-by-Step Solution

Answer

The simplified expression is \(6u^2 - 7uv - 10v^2\).

Step 1: Understand the FOIL Method

The FOIL method is a technique for multiplying two binomials. It stands for First, Outer, Inner, and Last, which are the pairs of terms from each binomial that need to be multiplied together.

Step 2: Multiply First Terms

Multiply the first terms of each binomial: \((6u) \times (u) = 6u^2\).

Step 3: Multiply Outer Terms

Multiply the outer terms of each binomial: \((6u) \times (-2v) = -12uv\).

Step 4: Multiply Inner Terms

Multiply the inner terms of each binomial: \((5v) \times (u) = 5uv\).

Step 5: Multiply Last Terms

Multiply the last terms of each binomial: \((5v) \times (-2v) = -10v^2\).

Step 6: Combine All Products

Combine all the products obtained in the previous steps: \(6u^2 - 12uv + 5uv - 10v^2\).

Step 7: Simplify the Expression

Combine like terms (the middle terms in this case): \(6u^2 - 7uv - 10v^2\).

Key Concepts

Multiplying Binomials

The FOIL method is a popular way to multiply binomials, specifically when dealing with expressions in the form of \((a+b)(c+d)\).This process involves four crucial steps:

• First Terms: Multiply the first terms of each binomial. In our case, it is \((6u) \times (u) = 6u^2\).
• Outer Terms: Multiply the outer terms. For our expression, this means \((6u) \times (-2v) = -12uv\).
• Inner Terms: Multiply the inner terms, resulting in \((5v) \times (u) = 5uv\).
• Last Terms: Multiply the last terms. Here, \((5v) \times (-2v) = -10v^2\).

These products are then combined to form the initial expanded expression: \(6u^2 - 12uv + 5uv - 10v^2\).This step sets the foundation for simplifying the expression.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations. They form the building blocks of algebra, allowing us to generalize mathematical concepts. In \((6u+5v)(u-2v)\), we see two binomial expressions being multiplied. Each binomial has two terms: a combination of a numeral coefficient and variables. For example, \(6u\) and \(5v\) in the first binomial.

The operation here is multiplication, which is simplified using the FOIL method. When multiplying algebraic expressions, the properties of operations, such as distributive, associative, and commutative properties, are key. They help rearrange and simplify terms, which is critical in combining terms later.

Simplifying Expressions

Simplifying expressions involves combining like terms to write an expression in its simplest form. This step makes the expression easier to work with and understand. After applying the FOIL method, we have \(6u^2 - 12uv + 5uv - 10v^2\).

To simplify:

• Identify like terms: Terms are considered 'like' if they have the same variable raised to the same power. In this expression, \(-12uv\) and \(5uv\) are like terms.
• Combine the coefficients of the like terms. \(-12uv + 5uv = -7uv\).

This leaves us with \(6u^2 - 7uv - 10v^2\), which is the simplest form of our original expression. Simplifying makes expressions more manageable and helps in solving equations or further manipulating the expression in more complex problems.