Problem 54 - Elementary Algebra

Question

For the following problems, find the products. $$ (y-7)(y+7) $$

Step-by-Step Solution

Answer

Answer: The product of the binomials (y-7)(y+7) is y^2 - 49.

Step 1: Apply the Distributive Property (FOIL Method)

First, let's recall what the FOIL method stands for: First, Outer, Inner, and Last. Multiply the terms in the following way: - First: Multiply the first term of the first binomial by the first term of the second binomial - Outer: Multiply the first term of the first binomial by the second term of the second binomial. - Inner: Multiply the second term of the first binomial by the first term of the second binomial. - Last: Multiply the second term of the first binomial by the second term of the second binomial.

Step 2: Carry out the Multiplications

Using the FOIL method, multiply the terms accordingly: - First: \(y \times y = y^2\) - Outer: \((y) \times (7) = 7y\) - Inner: \((-7) \times (y) = -7y\) - Last: \((-7) \times (7) = -49\)

Step 3: Combine Like Terms (if any) and Write the Final Result

Now, we need to combine any like terms (if there are any) and write the final result. In this case, we have two like terms: \(7y\) and \(-7y\). Combine these terms: \(7y - 7y = 0y\) We are left with: $$y^2 + 0y - 49$$ However, since 0y is equal to zero, we can simplify the result further: $$y^2 - 49$$ So, the product of the two given binomials is: $$ (y-7)(y+7) = y^2 - 49$$

Key Concepts

Distributive Property

The distributive property is a foundational aspect of algebra that allows us to multiply a single term by an entire group of terms inside parentheses. This property states that multiplying a sum (or difference) by a number will yield the same result as doing the multiplication for each addend separately and then adding (or subtracting) the products. In algebraic terms, for any numbers, a, b, and c, the distributive property can be represented as:
\[a(b + c) = ab + ac\]
In practice, when you come across a pair of binomials that need multiplying, such as \((y-7)(y+7)\), the distributive property enables you to 'distribute' the terms of the first binomial across the terms of the second binomial. This sets the stage for using the FOIL method to find the product of the binomials, ensuring that each term is accounted for in the final expression. This method will inevitably lead to the combination of like terms later in the process. Not only does the distributive property work for binomials, but it's also applicable to any polynomial, making it a versatile and essential tool in algebra.

Binomial Multiplication

Binomial multiplication involves multiplying two binomials together to obtain a single polynomial. A binomial is a polynomial with two terms, typically in the form \(a + b\) or \(a - b\). When it comes to multiplying binomials, the FOIL method acts as a systematic approach to ensure all parts of both binomials interact correctly. FOIL is an acronym for First, Outer, Inner, and Last, representing the order in which you should multiply the terms:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of each binomial.
• Inner: Multiply the inner terms of each binomial.
• Last: Multiply the last terms of each binomial.

By applying FOIL to the provided exercise, \((y-7)(y+7)\), each multiplication generates a specific product, producing a polynomial that may still contain terms that need to be simplified or combined. It's the consistency of this method that aids in learning and mastering more complicated multiplication as one progresses in algebra.

Combining Like Terms

After binomials are multiplied using the FOIL method, the next step is combining like terms—another crucial concept in algebra. Like terms are terms that have exactly the same variables raised to the same powers, though they may have different coefficients. For instance, in the exercise involving \((y-7)(y+7)\), after applying the FOIL method, the terms \(7y\) and \(-7y\) are like terms because they both contain the variable 'y' not raised to any power other than one.
In combining like terms, the goal is to simplify the expression by adding or subtracting the coefficients of these terms. In this particular exercise, \(7y\) and \(-7y\) cancel each other out, simplifying the expression to \(y^2 - 49\). This step is fundamental to algebra, as it allows for the simplification of an expression, which leads to clearer and often more concise results. Without this step of combining like terms, expressions would remain cluttered and difficult to work with in subsequent calculations. It's also a handy tool for checking the correctness of a solution by ensuring all possible simplifications have been made.