Problem 35 - Algebra 1: Concepts and Skills

Question

$$ (4 y-7)(2 y-1) $$

Step-by-Step Solution

Answer

The expanded form of the equation is \(8y^2 - 18y + 7\).

Step 1: Apply FOIL Rule

We have two binomials \(4y - 7\) and \(2y - 1\). First multiply the first terms of each binomial: i.e. \(4y * 2y\) which equals \(8y^2\).

Step 2: Multiply The Outside Terms

Multiply the outside terms of each binomial: i.e. \(4y * -1\) which equals \(-4y\).

Step 3: Multiply The Inside Terms

Multiply the inside terms of each binomial: i.e. \(-7 * 2y\) which equals \(-14y\).

Step 4: Multiply The Last Terms

Multiply the last terms of each binomial: i.e. \(-7 * -1\) which equals \(7\).

Step 5: Add All The Terms

Finally, add together these four terms to get the expanded form of the equation: \(8y^2 - 4y - 14y + 7\). This simplifies to \(8y^2 - 18y + 7\).

Key Concepts

FOIL Method

The FOIL method is a handy technique for multiplying two binomials. It helps to remember the order in which to multiply the terms: First, Outside, Inside, Last. This sequence ensures all terms in both binomials are multiplied. For example, given \((4y - 7)(2y - 1)\), the FOIL method guides us through:

• First: Multiply the first terms of each binomial. Here, that is \(4y \times 2y = 8y^2\).
• Outside: Next, multiply the outside terms: \(4y \times -1 = -4y\).
• Inside: Multiply the inside terms: \(-7 \times 2y = -14y\).
• Last: Finally, multiply the last terms: \(-7 \times -1 = 7\).

This systematic approach ensures all interactions between terms are accounted for, making it easier to find the full expanded expression.

Polynomial Expansion

Polynomial expansion involves multiplying expressions such as binomials to express them as a sum of terms. With the FOIL method, we expanded \((4y - 7)(2y - 1)\) to find: - \(8y^2 - 4y - 14y + 7\). The key to successful polynomial expansion is carefully performing each multiplication step and combining like terms in the final expression.

After multiplying and collecting all terms using the FOIL method, we proceed to combine similar terms for simplification. In our example: - We combined the terms \(-4y\) and \(-14y\) to get a single term \(-18y\).

Thus, the expanded form of the binomials becomes \(8y^2 - 18y + 7\). This process illustrates the beauty of polynomial expansion: transforming products into sums with simpler terms to understand and work with.

Algebraic Expressions

Algebraic expressions consist of variables and constants combined using arithmetic operations. Understanding how these expressions are formed and manipulated is crucial in algebra, especially when working with binomials.

In expressions such as \(4y - 7\) and \(2y - 1\), variables (\(y\)) and coefficients (numbers that multiply the variables) play vital roles. When these binomials are multiplied using the FOIL method, they form a new algebraic expression \(8y^2 - 18y + 7\).

Here are a few important points regarding algebraic expressions:

• The terms in an expression are separated by plus or minus signs.
• Each term can have a coefficient, variable, and an exponent.
• Simplifying expressions involves combining like terms (terms with the same variable and exponent).

Grasping the concept of algebraic expressions allows you to not only perform multiplications but also to simplify, evaluate, and understand algebra's broader applications.