Problem 59 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ a b c(3 a b c+c+b)+6 a\left(2 b c+b c^{2}\right) $$

Step-by-Step Solution

Answer

Question: Simplify the given algebraic expression: $$abc(3abc + c + b) + 6a(2bc + bc^2)$$ Answer: The simplified expression is $$3a^2b^2c^2 + ab^2c^2 + a^2b^2c + 12ab^2c + 6ab^3c^2$$.

Step 1: Distribute the terms inside the parenthesis

First, we need to distribute the terms for each part of the given expression: $$ a b c(3 a b c + c + b) + 6 a(2 b c + b c^2) $$

Step 2: Simplify each part of the expression

After distributing, we can simplify each part of the expression: $$ (3 a^2 b^2 c^2 + a b c^2 + a^2 b^2 c) + (12 a b c + 6 a b c^2) $$

Step 3: Combine like terms

Now, we will combine the like terms: $$ 3 a^2 b^2 c^2 + a b c^2 + a^2 b^2 c + 12 a b c + 6 a b c^2 $$ The expression is now simplified.

Key Concepts

Simplification

Simplification in algebra involves breaking down expressions into their simplest form. This can help make complex equations easier to understand and solve. The process entails several steps, such as removing parentheses, combining like terms, and ensuring the expression is as compact as possible.

When simplifying algebraic expressions, the goal is to reach a point where no further reduction is possible without changing the expression's value. This makes it more manageable for evaluation or further manipulation in solving equations. Consider it as tidying up a messy room, where items (terms) are grouped and organized efficiently.

Distributive Property

The distributive property is one of the foundational rules of algebra. It states that for any three numbers, say, \(a\), \(b\), and \(c\), the following equation holds: \(a(b + c) = ab + ac\). This property allows you to multiply a term across a sum or difference within parentheses, distributing the multiplication over each term separately.For example:

• If you have the expression \(3(x + y)\), you would use the distributive property to obtain \(3x + 3y\).
• In our original problem, we apply it to terms like \(a b c(3a b c + c + b)\), resulting in \(3a^2b^2c^2 + abc^2 + a^2b^2c\).

The distributive property simplifies expressions, making them easier to handle in subsequent steps of algebraic manipulation.

Combining Like Terms

Combining like terms is an essential part of simplifying algebraic expressions. Like terms are terms within an expression that have identical variables raised to the same powers. By adding or subtracting these terms, you consolidate them, reducing the expression's number of terms without altering its value.For instance:

• The terms \(3a\) and \(5a\) can be combined to form \(8a\).
• In our example, the expression \(a b c^2 + 6 a b c^2\) can be combined into \(7 a b c^2\).

This step is crucial to simplify and solve equations efficiently.

Polynomials

A polynomial is an algebraic expression made up of terms, which are composed of variables raised to whole-number exponents and coefficients. They are one of the most general forms of algebraic expressions you'll encounter in mathematics, from simple linear expressions to complex equations with multiple variables and terms.Polynomials come in various forms, such as:

• Monomials (a single term, e.g., \(5x\)).
• Binomials (two terms, e.g., \(x + 3\)).
• Trinomials (three terms, e.g., \(x^2 + 2x + 1\)).

The expression in the original exercise is a polynomial because it consists of multiple terms, each combining variables and coefficients.Understanding polynomials is crucial, as they are a foundational concept in algebra, and mastering them is key to solving many mathematical problems efficiently.