Problem 57 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 3 m\left[5+2 m\left(m+6 m^{2}\right)\right]+m\left(m^{2}+4 m+1\right) $$

Step-by-Step Solution

Answer

Question: Simplify the following algebraic expression: $$3 m\left[5+2 m\left(m+6 m^{2}\right)\right]+m\left(m^{2}+4 m+1\right)$$ Answer: The simplified expression is: $$36m^4 + 7m^3 + 4m^2 + 16m$$

Step 1: Distribute terms inside parentheses

We'll first distribute the terms inside the parentheses. $$ 3 m\left[5+2 m\left(m+6 m^{2}\right)\right]+m\left(m^{2}+4 m+1\right) = 3m(5) + 3m(2m^2+12m^3) + m^3 + 4m^2 + 1m $$

Step 2: Distribute further and simplify

Now distribute 3m to both terms and then simplify the expression: $$ 15m + 3m(2m^2+12m^3) + m^3 + 4m^2 + m = 15m + 6m^3 + 36m^4 + m^3 + 4m^2 + m $$

Step 3: Combine like terms

Finally, we will combine the like terms to reach the simplified expression: $$ 15m + 6m^3 + 36m^4 + m^3 + 4m^2 + m = 36m^4 + (6m^3 + m^3) + 4m^2 + (15m + m) $$ The simplified expression is: $$ 36m^4 + 7m^3 + 4m^2 + 16m $$

Key Concepts

Distributive Property

The distributive property is a key concept when simplifying algebraic expressions, especially when variables are involved. It's a way of distributing one term across other numbers or variables inside parentheses to simplify expressions. Think of it like sharing: you're spreading out a term across other elements it is multiplied by.
Here's how you apply it:

• When you see an expression like \(a(b + c)\), you distribute \(a\) to both \(b\) and \(c\), resulting in \(ab + ac\).
• This method helps break down complex expressions, making them easier to handle.

In our original problem, we use the distributive property to open up the expression \(3m[5 + 2m(m + 6m^2)]\). We distribute \(3m\) to every term inside the parentheses. So, \(3m\) is multiplied by \(5\) and \(2m(m + 6m^2)\). Then further in the sub-expression \(2m(m + 6m^2)\), \(2m\) is distributed over \(m\) and \(6m^2\). This method systematically reduces the complexity of the exercise.

Combining Like Terms

Combining like terms is another essential part of simplifying polynomial expressions. After distributing, expressions often have similar terms that can be combined to simplify further. Like terms are those with the same variable raised to the same power.
For example, in the expression \(5m^2 + 3m^2\), the terms \(5m^2\) and \(3m^2\) are like terms because both have the same variable \(m\) raised to the power of 2.

• To combine, simply add or subtract the coefficients of these terms.
• This makes the expression shorter and simpler.

In the original exercise, after using the distributive property, we arrive at a longer expression with various terms: \(15m + 6m^3 + 36m^4 + m^3 + 4m^2 + m\).
Through combining, \(6m^3\) and \(m^3\) combine to form \(7m^3\), while \(15m\) and \(m\) combine to form \(16m\). This process results in a tidier expression.

Polynomial Expressions

Understanding polynomial expressions is crucial, as they form the basis of many algebraic simplifications. A polynomial is a mathematical expression consisting of variables, coefficients, and the operations of addition, subtraction, and multiplication.
Key characteristics of polynomials include:

• The exponents of the variables are whole numbers.
• Polynomials are usually written in descending order of their degree.

An expression like \(36m^4 + 7m^3 + 4m^2 + 16m\) is a polynomial where the degrees range from 4 to 1. Each term is a part of the polynomial expression representing different powers of the variable \(m\). Simplifying a polynomial means combining like terms and ensuring there's no redundancy in the terms expressed.
This gives clarity and eases further calculations or evaluations. Understanding polynomials and how to work with them is a foundational skill in algebra, serving as a gateway to more advanced math topics.