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Chapter 10: Problem 446

Factor the greatest common factor from each polynomial. $$9 y-9$$

Short Answer

Expert verified
The factored form is $$9(y - 1)$$

Step by step solution

01

Identify the terms

Look at the given polynomial: $$9y - 9$$Identify the terms in the polynomial. Here, the terms are $$9y$$ and $$-9$$.
02

Find the Greatest Common Factor (GCF)

Determine the greatest common factor (GCF) of the coefficients and any common variables. Both terms have a coefficient of 9, so the GCF is 9.
03

Factor out the GCF

Factor out the GCF from each term in the polynomial: $$9y - 9 = 9(y - 1)$$
04

Verify the factorization

To make sure the factorization is correct, distribute the GCF back through the parentheses: $$9(y - 1) = 9y - 9$$

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Key Concepts

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

Factoring Polynomials
Factoring polynomials may sound complicated, but it's a process that simplifies expressions. This involves breaking down a polynomial into a product of simpler terms, known as factors. Here, our goal is to expose shared factors.

For the polynomial given: \(9y - 9\), we can break it into factors. Think of it like breaking a big thing into smaller, manageable pieces. We start by identifying common factors in each term.

Factoring helps in simplifying complex equations, solving polynomials, and understanding their properties. It鈥檚 an essential skill in algebra and beyond.
GCF (Greatest Common Factor)
The Greatest Common Factor (GCF) is the largest factor shared by two or more terms or numbers. In the context of polynomials, it simplifies the equation by dividing terms by the highest common factor.

For example, in \(9y - 9\), we first identify the GCF for 9. Since both terms (9y and -9) share 9 as a factor, the GCF is 9. Identifying the GCF makes factoring straightforward and effective.

Steps to find the GCF include:
  • List all factors of each term.
  • Identify the largest common factor.
  • Use the GCF to simplify the polynomial.
Recognizing the GCF is a key step in polynomial factorization.
Polynomial Factorization
Polynomial factorization refers to expressing a polynomial as a product of its factors. This process elucidates the structure of polynomials, making them easier to work with.

In the provided example, \(9y - 9\), after finding the GCF (which is 9), we can factor it out from each term:

\(9y - 9 = 9(y - 1)\)

Thus, \(9y - 9\) is factored into \(9(y - 1)\). Verifying helps ensure accuracy. Distribute the 9 back through the parentheses:

\(9(y - 1) = 9y - 9\)

Polynomial factorization simplifies solving, graphing, and analyzing polynomials efficiently.

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