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Chapter 7: Problem 26

Factor the greatest common factor from each polynomial. \(4 n-4\)

Short Answer

Expert verified
Factor: 4(n - 1).

Step by step solution

01

Identify the coefficients and constants

The terms in the polynomial are 4 and -4.
02

Find the Greatest Common Factor (GCF)

Determine the GCF of the coefficients 4 and -4. The GCF is 4.
03

Factor out the GCF from each term

Divide each term by the GCF and factor it out. Starting with the polynomial we get . The factorization is: 4(n - 1).

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Key Concepts

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

Greatest Common Factor
When factoring polynomials, the greatest common factor (GCF) is the largest factor that divides each term of the polynomial without leaving a remainder. In simpler terms, it's the highest number that you can evenly divide each term of the polynomial by.

Suppose you have the polynomial expression: 4n - 4. To factor this polynomial, first identify the GCF of the terms. The terms here are 4n and -4. The numbers (or coefficients) involved are 4 and -4.

The GCF of 4 and -4 is 4. Remember, the GCF must be a positive number even if some terms are negative.

To summarize, finding the GCF involves these steps:
  • Identify the coefficients of each term.
  • Determine the largest number that divides all coefficients evenly.
  • Remember that the GCF is always positive.
Identifying Coefficients
In a polynomial, coefficients are the numerical parts of the terms. They tell you how many times to multiply the variable. For example, in the expression 4n - 4:
  • The term 4n has a coefficient of 4.
  • The term -4 stands alone as a constant but can be thought of as -4 times n0, essentially having a coefficient of -4.


To identify the coefficients:
  • Look at each term in the polynomial.
  • For terms with variables, note the number in front of the variable (this is the coefficient).
  • For constant terms, the number itself is the coefficient.


Knowing the coefficients is a key part of finding the GCF and factoring the polynomial.
Factoring Process
Factoring is the process of breaking down a polynomial into products of other simpler polynomials. Here's the step-by-step process for our example: 4n - 4.

  • Step 1: Identify the coefficients and constants - In this polynomial, the terms are 4n and -4. Their coefficients are 4 and -4.
  • Step 2: Find the Greatest Common Factor (GCF) - We found that the GCF of 4 and -4 is 4.
  • Step 3: Factor out the GCF - This involves dividing each term by the GCF and then writing the polynomial as a product. For the polynomial 4n - 4, we divide both terms by 4 and factor out the GCF: 4(n) - 4(1), which simplifies to 4(n - 1).


The final factored form of the polynomial 4n - 4 is indeed 4(n - 1). This means we have expressed the original polynomial as a product of the GCF and the simplified form of the polynomial inside the parentheses.

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Most popular questions from this chapter

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