Problem 44 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using the FOIL method, and simplify. \((7 y-3)(2 y-1)\)

Step-by-Step Solution

Answer

The simplified expression is \(14y^2 - 13y + 3\).

Step 1: Apply the FOIL Method (First Terms)

Start by multiplying the first terms in each binomial. Multiply \(7y\) from the first binomial by \(2y\) from the second binomial: \(7y \times 2y = 14y^2\).

Step 2: Apply the FOIL Method (Outer Terms)

Next, multiply the outer terms. Multiply \(7y\) from the first binomial by \(-1\) from the second binomial: \(7y \times -1 = -7y\).

Step 3: Apply the FOIL Method (Inner Terms)

Then, multiply the inner terms. Multiply \(-3\) from the first binomial by \(2y\) from the second binomial: \(-3 \times 2y = -6y\).

Step 4: Apply the FOIL Method (Last Terms)

Finally, multiply the last terms. Multiply \(-3\) from the first binomial by \(-1\) from the second binomial: \(-3 \times -1 = 3\).

Step 5: Combine All the Products

Now, combine all these products together: \(14y^2 - 7y - 6y + 3\).

Step 6: Simplify the Expression

Combine like terms. The like terms here are \(-7y\) and \(-6y\). Adding them up gives: \(-13y\). The simplified expression is \(14y^2 - 13y + 3\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. They help us represent mathematical relationships and real-world problems in a symbolic form.
A key feature of algebraic expressions is their use of variables. Variables, often represented by letters like \( y \) or \( x \), allow flexibility in mathematical modeling. They stand in place of unknown or changeable quantities.
In expressions like \((7y - 3)(2y - 1)\), algebraic principles guide us in performing operations such as expansion and simplification.
Understanding the role of each component in an expression is critical for effectively applying algebraic techniques such as the FOIL method. Mastery of these elements builds a strong foundation for tackling more complex algebraic operations.

Binomials

Binomials are a specific type of algebraic expression composed of exactly two terms. These terms are typically separated by a plus or minus sign. For example, in the expression \((7y - 3)(2y - 1)\), both \(7y - 3\) and \(2y - 1\) are binomials.
The simplicity of binomials makes them excellent candidates for operations like multiplication and factoring, and they often occur in a variety of mathematical and real-life contexts.
When multiplying binomials, the FOIL method is particularly useful. This mnemonic helps remember the order of multiplication: First, Outer, Inner, Last. Notably, one multiplies the First terms (\(7y\) and \(2y\)), Outer terms (\(7y\) and \(-1\)), Inner terms (\(-3\) and \(2y\)), and Last terms (\(-3\) and \(-1\)).
Recognizing binomials and understanding their structure is essential in algebraic tasks, especially in polynomial operations.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials to produce another polynomial. In the case of binomials, this process often employs the FOIL method to ensure that all terms are effectively multiplied.
When multiplying binomials like \((7y - 3)(2y - 1)\), each term from the first binomial is multiplied by every term in the second, resulting in products like \(14y^2, -7y, -6y,\) and \(3\). These products represent the expanded form of the product of the binomials.
It's essential to keep track of signs during multiplication. For instance, multiplying negative numbers results in positive products, as seen in \(-3 \times -1 = 3\).
Understanding polynomial multiplication is crucial because it is a foundational skill for higher-level algebra, calculus, and various applications in science and engineering.

Simplification in Algebra

Simplification in algebra is the process of condensing expressions to their simplest form. This is achieved by combining like terms, which are terms that have the same variables raised to the same power.
For the expression resulting from the FOIL method \(14y^2 - 7y - 6y + 3\), simplification involves adding the like terms \(-7y\) and \(-6y\) to get \(-13y\).
The final simplified expression, \(14y^2 - 13y + 3\), represents the most condensed version of the product of the two binomials.
Simplification not only makes expressions easier to work with but also is crucial in solving equations. It allows us to focus on the core components of expressions, leading to more effective problem-solving and a clearer understanding of algebraic relationships.