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Chapter 5: Problem 12

Find the greatest common factor for each group of terms. $$70 x^{2}, 84 x, 42 x^{3}$$

Short Answer

Expert verified
The greatest common factor is 14x.

Step by step solution

01

- Determine the coefficients

Identify the coefficients of each term. The terms are 70, 84, and 42.
02

- Find the GCD of the coefficients

Find the greatest common divisor (GCD) of the coefficients. The GCD of 70, 84, and 42 can be found by listing the factors:- Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70- Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42The common factors are 1, 2, 7, and 14. The greatest common factor is 14.
03

- Determine the common variables

Identify the variable part common to all terms. The terms are: \(70x^{2}\), \(84x\), \(42x^{3}\). The common variable is \(x\), and the lowest power of \(x\) is \(x^{1}\).
04

- Multiply the GCD and the common variable

Multiply the greatest common divisor of the coefficients by the lowest power of the common variable: 14 * \(x^{1}\). This gives 14x.

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Key Concepts

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

Coefficients
In algebra, coefficients play a crucial role in determining the greatest common factor (GCF) of a group of terms. A coefficient is the numerical part of a term that contains both numbers and variables. For example, in the term 70x^2, the number 70 is the coefficient, while x^2 is the variable part. Understanding how to identify coefficients is fundamental because it sets the stage for finding the greatest common divisor (GCD) among them.
Greatest Common Divisor (GCD)
The greatest common divisor (GCD) of a set of numbers is the largest number that divides each of them without leaving a remainder. To find the GCD of coefficients, such as 70, 84, and 42, we list out their factors:

  • Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70
  • Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
  • Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

By identifying the common factors among these numbers, we find 1, 2, 7, and 14. The greatest common factor is the largest of these, which is 14. This process helps simplify terms in algebraic expressions.
Variable Part
The variable part of terms can also contribute to finding the greatest common factor. In the exercise, the terms include variables: 70x^2, 84x, 42x^3. To determine the GCF for the variable part, we identify the variable common to all terms, which in this case is x.

Next, we look at the exponent of each x in the terms:
  • 70x^2: exponent is 2
  • 84x: exponent is 1
  • 42x^3: exponent is 3

The smallest exponent common to all terms is x^1. Therefore, the variable part of the GCF is x^1. Combining this with the GCD of the coefficients (14), the overall GCF for these terms is 14x.

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