Problem 73 - Elementary Algebra

Question

(Section 4.6) Find the product. \((5 m-3)(2 m+3)\).

Step-by-Step Solution

Answer

Answer: The product of (5m - 3) and (2m + 3) is 10m^2 + 9m - 9.

Step 1: Apply the distributive property

We will multiply each term in the first expression by each term in the second expression and then sum the products. So, we have: $$(5m - 3)(2m + 3) = (5m)(2m) + (5m)(3) - (3)(2m) - (3)(3)$$

Step 2: Perform the multiplications

Now, we will multiply each term: $$(5m)(2m) = 10m^2$$ $$(5m)(3) = 15m$$ $$(3)(2m) = 6m$$ $$(3)(3) = 9$$ So, the expression becomes: $$10m^2 + 15m - 6m - 9$$

Step 3: Combine like terms

Now, we will combine the like terms (\(15m\) and \(-6m\)): $$10m^2 + 15m - 6m - 9 = 10m^2 + (15 - 6)m - 9$$ $$10m^2 + 9m - 9$$ So, the product of \((5 m-3)\) and \((2 m+3)\) is: $$10m^2 + 9m - 9$$

Key Concepts

Distributive Property

Understanding the distributive property is crucial when you're multiplying polynomials. It's like sharing cookies equally among friends, ensuring each term in one polynomial multiplies every term in the other polynomial. In our example, the distributive property is employed to blend the terms from \(5m - 3\) with each term of \(2m + 3\), like a handshake between each pair.

So, when we distribute \(5m\) across \(2m + 3\), and -3 across \(2m + 3\) as well, we're ensuring every term 'meets' and contributes to the final expression. This process is foundational in algebra, essentially utilizing the 'everyone gives to everyone' rule. By adhering to this property, multiplying the polynomials becomes a structured sequence of simpler multiplications.

Example of Distributive Property

When we expand \(5m\) to the second polynomial, we get \(5m \cdot 2m\) and \(5m \cdot 3\). Similarly, -3 is distributed to get \( -3 \cdot 2m\) and \( -3 \cdot 3\), leading us to our expanded algebraic expression before combining like terms.

Combining Like Terms

Once we've expanded our expression by using the distributive property, the next step is to simplify the polynomial by combining like terms. What does this mean? Think of it like grouping similar fruits together; apples with apples, bananas with bananas. In algebra, we combine terms that have the same variable raised to the same power. This step is essential for simplifying the expression and making it more understandable.

In the given exercise, after distributing and expanding the terms, we see that there are two 'm' terms: \(15m\) and \( -6m\). These like terms are 'friends' that we can put together. By combining the coefficients of these terms (15 and -6), we are essentially adding or subtracting the number of 'm's we have, resulting in a more streamlined expression. It's an important bookkeeping part of polynomial multiplication that leads to a cleaner, final expression.

Visualizing the Combination

Imagine having 15 'm' blocks and removing 6 of them. You're left with 9 'm' blocks which is represented as \(9m\) in our final simplified expression after combining like terms.

Algebraic Expressions

Polynomials are a type of algebraic expression, which are mathematical phrases that can include numbers, variables, and operating symbols. They don't have equal signs like equations do; think of them as unfinished stories waiting for further information to reach a conclusion.

Both \(5m - 3\) and \(2m + 3\), from our exercise, are algebraic expressions. Each polynomial has 'terms' which are the building blocks of these expressions. When these polynomials are multiplied, the distributive property and combining like terms are the tools used to write a new algebraic story. The final, simplified expression we find, \(10m^2 + 9m - 9\), tells a concise story about the relationship between the terms that originated from the product of two algebraic expressions.

The Role of Algebraic Expressions

The power of algebraic expressions lies in their ability to represent complex relationships through a symbolic language. The expressions can be altered, expanded, factored, and rearranged to reveal new relationships and insights into problems, which is the beauty of algebra.