Problem 88 - Elementary Algebra

Question

Either factor out the greatest common factor or factor by grouping. $$ c^{2}+c^{4}+c^{5}+c^{6} $$

Step-by-Step Solution

Answer

c^2(1 + c^2 + c^3 + c^4)

Step 1: Identify the Greatest Common Factor (GCF)

First, examine each term to identify if there is a common factor that can be factored out. The terms are: \( c^2, c^4, c^5, c^6 \). The greatest common factor in this case is \( c^2 \).

Step 2: Factor out the GCF

Factor \( c^2 \) out of each term: \[ c^2(c^0 + c^2 + c^3 + c^4) \]. Remember that \( c^0 = 1 \).

Step 3: Simplify the Expression

Rewrite the simplified terms inside the parentheses: \[ c^2(1 + c^2 + c^3 + c^4) \].

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is fundamental when factoring algebraic expressions. It refers to the highest degree of a variable or the largest number that evenly divides each of the terms in an expression.
To find the GCF:

• List the factors of each term.
• Identify the greatest common factor shared by all terms.

In the problem, the terms are: \( c^2, c^4, c^5, \text{ and } c^6 \). Here, the highest common factor is \( c^2 \) because it is the smallest exponent that divides all the exponents in the problem.

Factoring Polynomials

Factoring polynomials is the process of breaking down complex expressions into simpler parts. Factoring is critical as it simplifies solving equations and understanding the behavior of polynomials.
Here's a simple approach to factoring polynomials:

• Always start by finding the GCF and factor it out first.
• Next, check for patterns or common methods, like factoring by grouping or using special formulas for sums or differences of cubes or squares.

In our example, after identifying the GCF as \( c^2 \), we factored it out to get: \( c^2(c^0 + c^2 + c^3 + c^4) \).

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation signs. Understanding these expressions is crucial for algebraic manipulations like factoring and simplifying.
The terms in any algebraic expression are separated by plus or minus signs. For example, in the expression \( c^2 + c^4 + c^5 + c^6 \), each term (\( c^2 \), \( c^4 \), \( c^5 \), \( c^6 \)) consists of a variable raised to a power.
When working with algebraic expressions, always look for opportunities to combine like terms and factor out common factors.

Simplifying Expressions

Simplifying expressions involves making them more compact and easier to work with. This typically includes factoring, combining like terms, and reducing fractions. Simplification helps in solving algebra equations and inequalities effectively.
In simplified form, an expression should have the smallest number of terms and factors. For instance, starting from \( c^2 + c^4 + c^5 + c^6 \), we factored out the greatest common factor \( c^2 \) and simplified it to \( c^2(1 + c^2 + c^3 + c^4) \).
Always remember to check your work for errors and ensure that the simplified expression is correct and truly simplified.