Problem 34 - Algebra 2

Question

Simplify. $$ (4 y-6)(2 y+7) $$

Step-by-Step Solution

Answer

The simplified expression is \(8y^2 + 16y - 42\).

Step 1: Apply the Distributive Property

The first step in simplifying the expression is to apply the distributive property, also known as the FOIL method (First, Outer, Inner, Last) for binomials. Expand \((4y-6) \cdot (2y+7)\) as follows: - Multiply the First terms: \(4y \cdot 2y = 8y^2\) - Multiply the Outer terms: \(4y \cdot 7 = 28y\) - Multiply the Inner terms: \(-6 \cdot 2y = -12y\) - Multiply the Last terms: \(-6 \cdot 7 = -42\)

Step 2: Combine Like Terms

Now that we have expanded the expression, combine the like terms. The expanded version is as follows: \[8y^2 + 28y - 12y - 42\] Combine the terms with \(y\): \[28y - 12y = 16y\] Thus, the expression simplifies to: \[8y^2 + 16y - 42\]

Step 3: Final Step: Write the Simplified Expression

After combining like terms, write down the final simplified polynomial, which is: \[8y^2 + 16y - 42\] This is the fully simplified form of the original expression \((4y-6)(2y+7)\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps in simplifying expressions, especially when dealing with polynomials. It states that you can multiply a single term by terms inside a parenthesis independently and then add or subtract the results. In this exercise, the distributive property is used in the FOIL method to help expand the binomials \((4y - 6)(2y + 7)\) into a polynomial equation.

• First: Multiply the first terms of each binomial. For \(4y - 6\) and \(2y + 7\), this is \(4y \times 2y = 8y^2\).
• Outer: Multiply the outer terms in the products. Here, it's \(4y \times 7 = 28y\).
• Inner: Multiply the inner terms. That's \(-6 \times 2y = -12y\).
• Last: Multiply the last terms in the expression, \(-6 \times 7 = -42\).

Adding together the results, you get the expanded expression: \[8y^2 + 28y - 12y - 42\].The distributive property is crucial for transforming expressions into a form that can be further simplified.

Binomials

A binomial is a polynomial with two terms, often connected by a plus or minus sign, such as \(4y - 6\) or \(2y + 7\). Binomials are significant in algebra because they provide a basic structure for more complex expressions and are common in various algebraic operations and functions.
In the given exercise, two binomials are multiplied, demonstrating how their interaction results in a quadratic polynomial. The interaction of binomials is often tackled using the FOIL method, making it easier to handle operators and produce a simplified form. This process serves as a foundational step in many algebraic operations, preparing students for more advanced topics like trinomial expansion or polynomial factorization.
Understanding how to work with binomials offers a solid grounding for exploring polynomial equations, an essential part of algebra.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms that contain the same variable raised to the same power, which means they can be combined into a single term. In the exercise, after distributing, the terms with \(y\) were \(28y\) and \(-12y\).
To combine these like terms, simply add or subtract the coefficients, which are the numerical parts of the terms:

• Calculating \(28y - 12y\) results in \(16y\).

This step helps consolidate the expression from \[8y^2 + 28y - 12y - 42\] to its simplified form of \[8y^2 + 16y - 42\].
Combining like terms is a valuable technique that simplifies expressions and equations, making them easier to solve or understand. It’s a foundational skill in any algebraic procedure, enabling the transformation of complex-looking equations into more manageable forms.