Problem 126 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ c^{3}-3 c^{2}+9 c+3 c^{2}-9 c+27 $$

Step-by-Step Solution

Answer

The simplified expression is \( c^3 + 27 \).

Step 1: Combine like terms

Let's identify and combine the like terms from the polynomial expression. The terms involving the same power of \( c \) can be combined:1. Combine \( c^2 \) terms: \(-3c^2 + 3c^2 = 0\).2. Combine \( c \) terms: \( 9c - 9c = 0\).This reduces the expression to:\[ c^3 + 27 \]

Step 2: Simplified expression

After combining like terms, the simplified expression is formed. The polynomial has no more like terms to combine, leaving us with:\[ c^3 + 27 \]

Step 3: Finalize the expression

The expression can potentially be further simplified by recognizing any special algebraic forms. \( c^3 + 27 \) can be expressed as a sum of cubes since \( 27 = 3^3 \):\[ c^3 + 3^3 \]However, in this scenario, this is the simplified form as a sum of cubes indicates it can't be simplified further without specific factorization called sum of cubes.

Key Concepts

Polynomial Expressions

Polynomial expressions are expressions made up of terms that are added or subtracted. These terms must have variables raised to whole number exponents.
Each term in a polynomial is essentially a product of a coefficient (a number) and a power of the variable. In the expression given, terms like \(c^3\), \(-3c^2\), and \(9c\) are all parts of a polynomial expression because they involve powers of the variable \(c\).

• Terms like \(c^3\) are known as monomials since they are singular parts of the polynomial.
• The degree of a polynomial is determined by the highest power of the variable present; in this case, it is 3 because of \(c^3\).
• Coefficients are the numbers in front of the variables, such as -3 in \(-3c^2\).

Understanding the structure of polynomial expressions is essential for further simplification processes, such as combining like terms and recognizing special forms like the sum of cubes.

Combining Like Terms

Combining like terms is a fundamental process in algebra that involves simplifying expressions by merging identical variable terms.
Terms are considered "like" if they have the same variable raised to the same power. For instance, the terms \(-3c^2\) and \(3c^2\) are like terms because they both feature \(c\) raised to the power of 2.
When combining like terms, consider:

• Variable and Exponent: Ensure the variable and its exponent are identical.
• Add or Subtract Coefficients: Adjust the coefficients accordingly. For example, \(-3c^2 + 3c^2 = 0\). This results in the terms canceling each other out.
• Simplification: The objective is to reduce the expression, decreasing the number of terms.

By effectively combining like terms, as shown in the exercise, the expression \(c^3 - 3c^2 + 9c + 3c^2 - 9c + 27\) simplifies neatly to \(c^3 + 27\). This step makes solving and understanding polynomial expressions much clearer.

Sum of Cubes

The sum of cubes is an important algebraic identity that involves expressing a sum of two cube terms. In our case, the final simplified expression \(c^3 + 27\) can be written as a sum of cubes because \(27\) equals \(3^3\).
This particular identity is typically expressed in the form \(a^3 + b^3\), which can be factored as \((a + b)(a^2 - ab + b^2)\).

• Recognition: To identify a sum of cubes, check if each term is a cube. Here, \(c^3\) and \(3^3\) confirm this qualification.
• Factorization: Although \(c^3 + 27 = c^3 + 3^3\) is already a sum of cubes, it can be factored using the formula, but unless needed, it is often left as is.
• Application: Understanding when and how to use the sum of cubes aids in more advanced algebraic manipulations and problem-solving scenarios.

In many contexts, recognizing a sum of cubes is enough for simplification without extensive factorization, especially in basic algebra exercises.