Problem 82 - Elementary Algebra

Question

\(9 a\left(8 a^{2}\right)+9 a(4 a)+9 a(6)\)

Step-by-Step Solution

Answer

9a (8a^2 + 4a + 6)

Step 1: Factor Out the Common Term

Notice that each term in the expression has a common factor of 9a. Factor 9a out from each term inside the parentheses: \[ 9a (8a^2 + 4a + 6) \]

Step 2: Simplify Inside the Parentheses

After factoring out 9a, simplify the expression inside the parentheses by combining like terms, if any. Here, the terms inside \(8a^2 + 4a + 6\) are already simplified and cannot be combined further.

Step 3: Final Simplified Expression

The final expression after factoring out 9a is:\[ 9a (8a^2 + 4a + 6) \]

Key Concepts

Common Factors

When dealing with algebraic expressions, a common factor refers to a number or variable that is present in each term of the expression. Identifying and factoring out common factors helps simplify expressions.
In our exercise, the common factor is 9a.
This can be seen because each term includes this factor:

• First term: 9a × 8a³
• Second term: 9a × 4a
• Third term: 9a × 6

Factoring out 9a from the entire expression consolidates it, making it easier to handle. The importance of identifying common factors cannot be overstated, as it streamlines problem-solving and lays the groundwork for further simplifications.

Simplifying Expressions

Simplifying an expression involves reducing it into its simplest form. This means performing operations and combining like terms, where possible. By simplifying, we make the expression easier to work with.
For our exercise: 1. First, identify the common factor, which we've done as 9a. 2. Factor out 9a to get: \(9 a (8 a^{2} + 4 a + 6)\) Each term inside the parentheses, \(8a^2 + 4a + 6\), does not have any like terms that can be combined or simplified further. This step highlights that sometimes, simplification may involve only factoring without further reduction inside the parentheses. This concept reiterates the core of algebra: reviewing each term and handling them systematically.

Polynomials

A polynomial is an expression made up of variables (also called indeterminates) and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

• The polynomial in this exercise is: \(8a^2 + 4a + 6\).

Polynomials can be of varying degrees, determined by the highest power of the variable. Here, the term \(8a^2\) has the highest power, making our polynomial a quadratic one (because of the power 2). Understanding polynomials is fundamental since they form the basis of algebraic equations, and knowing how to manipulate them, like factoring and simplifying, is essential for solving more complex math problems.
Keep practicing, and soon these concepts will become second nature!