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Chapter 10: Problem 477

Factor the greatest common factor from each polynomial. $$12 u^{2}-36 u-108$$

Short Answer

Expert verified
12(u^2 - 3u - 9)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, find the greatest common factor (GCF) of the terms in the polynomial. The terms are 12, -36, and -108. The GCF of these numbers is 12.
02

Factor Out the GCF

Next, factor out the GCF from each term in the polynomial: \[12u^2 - 36u - 108 = 12(u^2 - 3u - 9)\]

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Key Concepts

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

Greatest Common Factor (GCF)
The greatest common factor (GCF) is the largest number that divides all the terms in a polynomial without leaving a remainder. When factoring a polynomial, identifying the GCF is an essential first step.
In the exercise, the terms of the polynomial are 12, -36, and -108. To find the GCF of these terms, you can list the factors of each number:
  • 12: 1, 2, 3, 4, 6, 12
  • -36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  • -108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
From these lists, the largest common factor is 12. Therefore, 12 is the GCF. Using the GCF helps simplify the polynomial, making further factorization more manageable.
Polynomial Factorization
Polynomial factorization involves expressing a polynomial as a product of its factors. These can include numbers, variables, or other polynomials. After finding the GCF, the next step is to factor it out.
In the given polynomial, \[12u^2 - 36u - 108\], we identified 12 as the GCF. Factoring out 12, we write the polynomial as: \[12(u^2 - 3u - 9)\]
This means we divide each term in the polynomial by 12, simplifying the expression inside the parentheses. The resulting polynomial inside the parentheses is now more manageable for further factorization if needed.
Overall, polynomial factorization simplifies complex expressions and is crucial in solving polynomial equations.
Prealgebra
Prealgebra lays the foundation for understanding basic algebra concepts, including factoring polynomials. When working with polynomials in prealgebra, you'll often begin with these key steps:
  • Identifying and understanding the terms of the polynomial
  • Finding the GCF of the polynomial terms
  • Factoring out the GCF to simplify the expression
By mastering prealgebra concepts like greatest common factor and polynomial factorization, you'll be better prepared to handle more complex algebra problems in the future.
Practicing these skills early on helps you develop a strong mathematical foundation.

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Most popular questions from this chapter

Multiply and write your answer in decimal form. $$\frac{6 \times 10^{9}}{2 \times 10^{-1}}$$

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Convert \(5.25 \times 10^{-4}\) to decimal form.

Divide the monomials. $$\frac{\left(5 m^{9} n^{3}\right)\left(8 m^{3} n^{2}\right)}{\left(10 m n^{4}\right)\left(m^{2} n^{5}\right)}$$

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