Problem 90 - Elementary Algebra

Question

How many \((2 a+5)\) 's are there in \(3 x(2 a+5)\) ?

Step-by-Step Solution

Answer

Answer: There are 0 (2a+5) terms in the expression 3x(2a+5).

Step 1: Expand the expression

Multiply \(3x\) by \((2a+5)\) to expand the expression: \(3x(2a+5) = 3x \cdot 2a + 3x \cdot 5 = 6ax + 15x\)

Step 2: Identify the number of \((2a+5)\) terms

We can see that there is no \((2a+5)\) factor in the simplified expression, \(6ax + 15x\). Hence, there are no \((2a+5)\) 's in \(3x(2a+5)\).

Key Concepts

Multiplication of Binomials

Multiplication of binomials is a fundamental concept in algebra. A binomial is an expression with two terms, such as \(2a + 5\). In this exercise, we are multiplying a monomial \(3x\) by a binomial \(2a + 5\). To solve these types of problems, we use techniques like the distributive property to simplify the expression. It is important to apply each term in the monomial to every term in the binomial.
For instance, when you multiply a monomial by a binomial, you distribute the monomial across each term within the binomial. This means you perform two separate multiplications:
\[3x \, \cdot \, 2a\]
and
\[3x \, \cdot \, 5\].
Once these calculations are done, you then add the results.

• Calculate \(3x \, \cdot \, 2a = 6ax\).
• Calculate \(3x \, \cdot \, 5 = 15x\).
• Finally, add them together to get \(6ax + 15x\).

This demonstrates how multiplying a monomial with a binomial can expand and simplify the original expression.

Distributive Property

The distributive property is a key mathematical principle used in algebra to break down expressions into simpler parts and multiply more efficiently. It essentially means distributing one operation across the terms of another.
In simpler terms, if you have a term \(A\) that multiplies a group of terms \(B + C\), you distribute \(A\) across \(B\) and \(C\). The expression \(A(B + C)\) becomes \(AB + AC\).
This property was used in our exercise when multiplying the monomial \(3x\) by the binomial \(2a + 5\).

• The first distribution was \(3x \, \cdot \, 2a = 6ax\).
• The second distribution was \(3x \, \cdot \, 5 = 15x\).

By applying the distributive property, you break down complex multiplication into simpler, manageable parts and see how each piece contributes to the overall solution.

Simplifying Expressions

Simplifying expressions is an integral part of solving algebraic problems. After applying multiplication and distributive techniques, what you often have is a longer expression that can be tough to understand. This is where simplification comes in.
The goal of simplifying is to reduce the expression to its most concise form without changing its value. This can include combining like terms and reducing the expression as much as possible. In our example, the initial multiplication resulted in \(6ax + 15x\), an expression that is expanded but can’t be reduced further by combining terms.
Why? Because there are no like terms—that is, terms that have the same variables raised to the same powers—within \(6ax + 15x\). Simplifying does not always imply reducing the number of terms; it involves making the expression as clear and uncomplicated as possible. Understanding when an expression is simplified enough is crucial to solving algebraic tasks efficiently.