Problem 34 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(6 x+5)(x+3)$$

Step-by-Step Solution

Answer

The product is \(6x^2 + 23x + 15\).

Step 1: Use the Distributive Property

First, apply the distributive property, which states that you multiply each term in the first binomial by each term in the second binomial: \((6x+5)(x+3) = 6x(x) + 6x(3) + 5(x) + 5(3)\).

Step 2: Multiply the Terms

Carry out each multiplication from Step 1: - \(6x \cdot x = 6x^2\) - \(6x \cdot 3 = 18x\) - \(5 \cdot x = 5x\) - \(5 \cdot 3 = 15\).

Step 3: Combine Like Terms

Add the products obtained in Step 2: \(6x^2 + 18x + 5x + 15\). Combine the like terms (i.e., terms with the same degree): \(6x^2 + (18x + 5x) + 15 = 6x^2 + 23x + 15\).

Key Concepts

Multiplying Binomials

Multiplying binomials involves the use of a property in algebra known as the distributive property. This property means you need to distribute each term of one binomial to every term of another binomial. Imagine two teams where each player plays against every player from the other team. For the binomials \((6x + 5)\) and \((x + 3)\), each term in the first binomial multiplies with each term in the second one.

Here’s how it breaks down:

• First, multiply the first term of the first binomial (\(6x\)) with both terms in the second binomial: \(6x \cdot x\) and \(6x \cdot 3\).
• Next, do the same with the second term of the first binomial (\(5\)): \(5 \cdot x\) and \(5 \cdot 3\).

This method ensures that all combinations of products from the two binomials are accounted for.

Ultimately, multiplying binomials helps us simplify quadratic expressions, which are crucial in understanding more complex algebraic problems.

Algebraic Expressions

Algebraic expressions can be seen as mathematical statements made up of numbers, variables, and operation signs. In our case, \((6x + 5)(x + 3)\) is an expression set to be expanded into a bigger picture. The goal is to handle the expression systematically:

• Recognize the variables (like \(x\)) as placeholders that can take any value.
• Identify coefficients, like \(6\) and \(5\), which multiply the variables, impacting their effect in equations.

Whenever you deal with expressions involving binomials, remember that you're exploring relationships between numbers and variables.

In algebra, learning to expand and simplify these expressions helps lay the groundwork for solving equations, allowing you to manipulate and break down complex problems into manageable steps.

Combining Like Terms

Combining like terms is an essential step in simplifying expressions. After you multiply binomials, you'll gather terms that are similar, meaning they have the same variable raised to the same power. This makes it easier to reduce the expression to its most succinct form.

In the expanded expression \(6x^2 + 18x + 5x + 15\), notice the terms \(18x\) and \(5x\). Both are like terms because they contain the variable \(x\) raised to the first power.

• Add these coefficients together (\(18 + 5\)) to get \(23x\).
• Leave other distinct terms like \(6x^2\) and constants like \(15\) unchanged.

The final simplified expression is \(6x^2 + 23x + 15\).

Mastering this skill is crucial, as it allows you to see how different parts of an expression relate, making equations much easier to work with and solve.