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Chapter 4: Problem 8

Factor each monomial. $$4 c^{2}$$

Short Answer

Expert verified
The factored form of \(4c^2\) is \((2 \times 2) \times (c \times c)\).

Step by step solution

01

Identify the Coefficient

The monomial provided is \(4c^2\). First, identify that the coefficient of this term is \(4\).
02

Prime Factorization of the Coefficient

Next, perform the prime factorization of \(4\). This gives \(4 = 2 \times 2\).
03

Identify the Variable Component

The variable part of the monomial is \(c^2\), which means "\(c\) raised to the power of 2".
04

Write Each Component in Factored Form

Write the factored form of the monomial using its components: \(c^2 = c \times c\). Therefore, \(4c^2 = (2 \times 2) \times (c \times c)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficients
When dealing with monomials, understanding coefficients is key. A coefficient is a number that multiplies a variable within the term. For the term \(4c^2\), the coefficient is \(4\). This number is not dependent on the variable. It stands alone when identified in the monomial.
  • The coefficient tells us how many times the variable part is used or considered for multiplication.
  • It's always a constant number and helps to scale or adjust the value of the variable part.
By distinguishing the coefficient, we can better factorize the entire monomial.
Prime Factorization
Prime factorization is breaking down a number into a product of its prime numbers. For the coefficient \(4\), this means finding prime numbers that multiply to form \(4\).
  • The prime factorization of \(4\) is \(2 \times 2\), as \(2\) is a prime number.
  • Prime numbers are numbers greater than \(1\) with no divisors other than \(1\) and themselves.
Prime factorization helps simplify the coefficient, especially when finding common factors or further simplifying expressions.
Variables
Variables are symbols representing numbers or values that can change within a mathematical expression. In the monomial \(4c^2\), \(c\) represents the variable.
  • Variables can take any value which is decided in the context of the problem.
  • They are often used to make generalizations in mathematical equations or expressions.
In this exercise, we see the variable part as \(c^2\), indicating how variables work with exponents to scale or reduce their effects.
Exponents
Exponents tell us how many times a number or variable is multiplied by itself. In \(4c^2\), the \(^2\) is the exponent applying to the variable \(c\).
  • An exponent of \(2\) means the variable is multiplied by itself: \(c \times c\).
  • Exponents can simplify repeated multiplication, making expressions shorter and easier to read.
When factoring, understanding exponents is crucial because they help determine how many times the base variable appears in the expression.

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