Problem 58 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 4 m(2 m+7) $$

Step-by-Step Solution

Answer

Question: Multiply the monomial 4m by the binomial 2m + 7, and then simplify the resulting expression. Answer: The simplified expression after multiplying the monomial by the binomial is 8m^2 + 28m.

Step 1: Distribute the monomial over the binomial

To multiply 4m by (2m + 7), distribute the 4m across both terms of the binomial by performing the multiplication: $$ 4m(2m + 7) = (4m \times 2m) + (4m \times 7) $$

Step 2: Perform the multiplications

Now, multiply the terms together: $$ (4m \times 2m) + (4m \times 7) = 8m^2 + 28m $$

Step 3: Combine like terms

In this case, there are no like terms to combine. The final simplified expression is: $$ 8m^2 + 28m $$

Key Concepts

Understanding the Distributive Property

The Distributive Property is a fundamental tool in algebra that allows you to simplify expressions and solve equations by distributing a multiplication operation across terms within parentheses. It states that if you have a term outside of parentheses, like in \( a(b+c) \), you can distribute \( a \) over each term inside the parentheses separately to get \( ab + ac \). This property is especially useful when dealing with polynomials.
In the given exercise, we apply the Distributive Property to multiply the monomial \( 4m \) with each term within the binomial \( (2m + 7) \). This operation simplifies the expression to \( (4m \times 2m) + (4m \times 7) \).
This step illuminates the distributive tactic across two separate terms, which forms the basis of polynomial expansion.
Remember, applying the Distributive Property correctly can make complex algebraic expressions much easier to manage and understand.

Defining a Binomial

In algebra, a binomial is simply an expression that contains exactly two distinct terms. These terms are usually joined by a plus \((+)\) or minus \((- )\) sign. The term 'binomial' comes from the prefix "bi-" which means "two."
For example, the expression \( (2m + 7) \) in the exercise is a binomial because it contains two terms (\(2m\) and \(7\)) that are added together. Binomials are an essential part of many algebraic operations, including polynomial multiplication.
When working with binomials, understanding each component is key. You'll notice each term can have variables, constants, or both. Knowing which is which helps simplify the multiplying and combining process.

• The first term (or leading term) in a binomial is often accompanied by a variable, as seen in \((2m + 7)\).
• The second term is typically a constant, such as the \(+ 7\) here.

Recognizing what constitutes a binomial is vital for applying the correct multiplication techniques and simplifying expressions in algebra.

Polynomial Multiplication Basics

Polynomial multiplication involves multiplying two polynomials to get another polynomial. When you multiply, each term in the first polynomial is distributed across all terms in the second polynomial. For this reason, understanding the systematic method is critical.
In this exercise, we focused on the multiplication of a monomial \((4m)\) and a binomial \((2m + 7)\). The process began by applying the Distributive Property, allowing us to multiply \(4m\) by each term in the binomial separately:

• \(4m \times 2m = 8m^2\)
• \(4m \times 7 = 28m\)

After performing these multiplications, we combine the results to get \(8m^2 + 28m\).
In general, when multiplying polynomials:

• Use the Distributive Property to tackle each term.
• Perform multiplications carefully, attending to variable exponents (add them when they share the same base).
• Combine like terms, which are terms with the same variable raised to the same power.

Mastering these steps will ensure you're prepared to handle various polynomial multiplication problems efficiently.