Problem 6 - Elementary Algebra

Question

In the expression \(10+2(b+6)(b-18)^{2},\) list the factors of the first term: second term:

Step-by-Step Solution

Answer

Answer: The factors of the expression 10 are 1, 10, 2, and 5. The factors of the expression 2(b+6)(b-18)^{2} are 2, (b+6), and (b-18)^{2}.

Step 1: Identify the factors of the given term.

We need to identify the factors of the first term, which is 10. The factors of 10 are the numbers that can multiply together to give the result of 10.

Step 2: List the factors.

After analyzing the given term, we can conclude the following factors for the first term (10): 1 x 10 and 2 x 5 For the second term:

Step 1: Identify the factors of the given term.

We need to identify the factors of the second term: 2(b+6)(b-18)^{2}. The factors of this term are the constant "2", the expression "(b+6)" involving the variable "b", and the expression "(b-18)^{2}" which also involves the variable "b".

Step 2: Write the expression in terms of its factors.

The second term can be written as the product of its factors: 2 * (b+6) * (b-18)^{2} First term factors: 1, 10, 2, 5 Second term factors: 2, (b+6), (b-18)^{2}

Key Concepts

Identifying Factors

Breaking down complex expressions into simpler multipliers is the essence of identifying factors in algebra. It's akin to dismantling a machine into its building blocks to understand its construction and function.

Let's take the example of the expression with the first term being 10. To identify its factors, we look for whole numbers that can be multiplied to yield 10. It’s similar to finding what ingredients and their respective quantities are needed to create a particular dish. In this case, the 'recipe' allows combining 1 and 10 or 2 and 5 to get the final product, 10. Thus, 1, 2, 5, and 10 are all factors of 10.

In algebra, this process is crucial since it simplifies expressions for solving or further manipulation and is especially beneficial when working with polynomials and equations.

Polynomial Expressions

Polynomial expressions are like algebra's sentences, forming the language of diverse mathematical scenarios. They consist of variables raised to whole number exponents, coefficients, and the operations of addition, subtraction, and multiplication.

An example of a polynomial expression is the second term in our given expression, termed as: \(2(b+6)(b-18)^2\). This term illustrates a polynomial with several components: a constant multiplier, 2; a linear term, \((b+6)\); and a squared term, \((b-18)^2\). Understanding the structure and being able to identify the individual polynomial members—the factors—lays the groundwork for operations such as expansion, simplification, and even finding roots or solving equations.

Polynomials can be as simple as a monomial or as complex as a lengthy combination of terms. They are the building blocks for many algebraic problems, serving as an essential skill for anyone venturing into algebraic studies or applications.

Algebraic Terms

Algebraic terms are the symbolic representatives of numbers in algebra, comprising constants or variables and incorporating the four basic operations of arithmetic, along with exponentiation.

The expression \((b+6)\) and \((b-18)^2\) from our second term are algebraic terms that include both constants and variables, where 'b' is the variable, 6 and 18 are constants. When variables are involved, terms represent an array of possible values, depending on the variable's value. A single algebraic term can have profound implications, determining the shape of a graph, the solution to an equation, or the growth of investment over time.

By breaking down and understanding each part of the second term, algebraic terms become the tools used to craft further mathematical expressions or solve algebraic puzzles. The power of algebra lies in the manipulation of these terms to unlock the mysteries presented in mathematical statements.