Problem 59 - Elementary Algebra

Question

For the following problems, find the products. $$ (5 h-2 k)(5 h+2 k) $$

Step-by-Step Solution

Answer

Question: Find the product of the given binomials: \((5h - 2k)(5h + 2k)\). Answer: The product of the given binomials is \((5h - 2k)(5h + 2k) = 25h^2 - 4k^2\).

Step 1: Identify the binomials

We are given the following binomials: \((5h - 2k)(5h + 2k)\)

Step 2: Apply the FOIL method

Using the FOIL method (First, Outer, Inner, Last), we will multiply each term in the first binomial by each term in the second binomial. First: \((5h)(5h)\) Outer: \((5h)(2k)\) Inner: \((-2k)(5h)\) Last: \((-2k)(2k)\)

Step 3: Multiply the terms

Now, we will multiply the terms we obtained in Step 2: First: \((5h)(5h) = 25h^2\) Outer: \((5h)(2k) = 10hk\) Inner: \((-2k)(5h) = -10hk\) Last: \((-2k)(2k) = -4k^2\)

Step 4: Combine like terms

Combine like terms by adding the coefficients of the terms that are alike. \(25h^2 + 10hk - 10hk - 4k^2\) The \(10hk\) and \(-10hk\) terms cancel each other out: \(25h^2 - 4k^2\)

Step 5: Write the final answer

The product of the given binomials is: $$(5h - 2k)(5h + 2k) = 25h^2 - 4k^2$$

Key Concepts

Binomial Product

Understanding the binomial product is essential when working with algebraic expressions. A binomial is a polynomial with exactly two terms, such as \(5h-2k\). When we multiply two binomials, we often use the FOIL method, which stands for First, Outer, Inner, Last. This methodical approach ensures that each term from one binomial multiplies by each term of the other.

The product of two binomials will be another polynomial. Importantly, the initial terms of each binomial (the 'First' terms) multiply together, as well as the 'Outer' terms, the 'Inner' terms, and the 'Last' terms of each binomial. Our aim is to not miss any terms and maintain the order of operations, which is crucial for the correct outcome.

For example, multiplying \(5h - 2k\) and \(5h + 2k\) involves expanding \(5h \cdot 5h\), \(5h \cdot 2k\), \( -2k \cdot 5h\), and \( -2k \cdot 2k\), reflecting the FOIL sequence. The clarity of this method allows us to systematically approach multiplication and obtain the precise product of two binomials.

Multiplying Polynomials

When it comes to multiplying polynomials, the principles are just an extension of what we use for binomials. However, as polynomials can have more than two terms, the process can involve more combinations. Multiplying each term in the first polynomial with every term in the second polynomial is critical to finding the correct product.

For instance, when dealing with binomials, the FOIL method is perfect. But for polynomials with three or more terms, we employ the distribute property (or distributive law), which states that \(a(b+c) = ab + ac\).

Why is this important?

Simply because it helps us keep track of all the terms and ensures no term is left behind. Properly expanding the expressions and being systematic in our approach prevents errors and confusion, leading to the exact expansion of polynomial products.

After distributing each term across the polynomial, we must align similar terms to prepare for the simplification step, which brings us to our next core concept: combining like terms.

Combine Like Terms

Combining like terms is a fundamental part of algebra that simplifies expressions and makes them easier to interpret. Like terms are the elements within an algebraic expression that have the same variables raised to the same power. Only the coefficients of these like terms are different.

After multiplying out the terms in polynomials or binomials, we often end up with some terms that can be consolidated. This is where we 'combine like terms' by adding or subtracting the coefficients and keeping the variable parts intact.

Simplification is Key

For our example \(25h^2 + 10hk - 10hk - 4k^2\), we see that \(10hk\) and \( -10hk\) are like terms, which means they cancel each other out because they have the same variable part but opposite coefficients. The simplified form of the expression, therefore, would only include the distinct terms, in this case \(25h^2\) and \( -4k^2\), giving us the final answer \(25h^2 - 4k^2\).