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Chapter 6: Problem 119

Factor. $$ 12 u v w^{3}-54 u v^{2} w^{2} $$

Short Answer

Expert verified
The factored form is \(6uvw^2(2w - 9v)\).

Step by step solution

01

Identify Common Factors

Identify common factors in the terms of the expression. The terms are 12uvw^3 and -54uv^2w^2. Both terms have the common factors of 6, u, v, and w^2.
02

Factor Out the Greatest Common Factor

Factor out the greatest common factor (GCF) from the expression. The GCF is 6uvw^2. Divide each term by the GCF. This results in: \[ 12uvw^3 - 54uv^2w^2 = 6uvw^2(2w - 9v) \]

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Key Concepts

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

Greatest Common Factor
The greatest common factor (GCF) is the largest factor that divides two or more numbers. When factoring polynomial expressions, finding the GCF helps simplify expressions by factoring out this common factor from all terms. This process simplifies equations and expressions, making them easier to work with.
To identify the GCF in the expression \(12uvw^3 - 54uv^2w^2\), first list out the factors:
  • For \(12uvw^3\): the factors are \(12\), \(u\), \(v\), and \(w^3\).
  • For \(-54uv^2w^2\): the factors are \(54\), \(u\), \(v^2\), and \(w^2\).
Next, find the common factors:
  • The numbers \(12\) and \(54\) have a common factor of \(6\).
  • Both terms share \(u\) and \(v\) as factors.
  • The smallest power of \(w\) is \(w^2\).
Thus, the GCF is \(6uvw^2\). Using the GCF simplifies expressions by reducing each term by this factor.
Polynomial Expressions
Polynomial expressions involve multiple terms grouped together using the operations of addition, subtraction, and multiplication. In the given problem, the polynomial expression \(12uvw^3 - 54uv^2w^2\) consists of two terms. Each term combines constants, variables, and exponents.
Understanding polynomial expressions entails recognizing their components:
  • **Constants**: These are fixed values, such as \(12\) and \(54\) in the expression.
  • **Variables**: Symbols like \(u\), \(v\), and \(w\) that represent numbers.
  • **Exponents**: Indicate how many times to multiply a variable by itself. For example, \(w^3\) means \(w\) multiplied by itself three times.
Recognizing these components allows you to identify common factors and perform operations like factoring, which simplify expressions and make it easier to solve equations.
Distributive Property
The distributive property is a fundamental principle in algebra used to simplify expressions and solve equations. This property states that multiplying a factor by a sum or difference inside parentheses is equal to multiplying the factor by each term separately, then adding or subtracting the resulting products.
In the expression \(12uvw^3 - 54uv^2w^2\), after identifying the GCF as \(6uvw^2\), you apply the distributive property:
  • First, divide each term by the GCF \(6uvw^2\).
  • This simplifies \(12uvw^3\) to \(2w\) and \(-54uv^2w^2\) to \(-9v\).
Hence, the expression becomes \(6uvw^2(2w - 9v)\).
This application shows the power of the distributive property in rewriting complex expressions into more manageable forms, helping solve algebraic equations efficiently.

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Most popular questions from this chapter

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ a^{3}-24-4 a+6 a^{2} $$

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 5 x^{3} y^{3} z^{4}+25 x^{2} y^{4} z^{2}-35 x^{3} y^{2} z^{5} $$

For each of the following polynomials, which factoring method would you use first? $$ m^{2}+3 m n+2 n^{2} $$

For each of the following polynomials, which factoring method would you use first? $$ 16-25 z^{2} $$

For each of the following polynomials, which factoring method would you use first? $$ a x+a y-x-y $$
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