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Chapter 6: Problem 103

Explain how to find the greatest common factor of a list of terms. Give an example with your explanation.

Short Answer

Expert verified
The greatest common factor (GCF) of the list [12, 18, 30] is 6

Step by step solution

01

Understanding the Problem

We need to understand that finding the GCF involves determining the largest number that evenly divides all the numbers in a list. For instance, if we have the list [12, 18, 30], the GCF is the biggest number that can divide 12, 18, and 30 with no remainder.
02

Identifying Prime Factors

Next, we need to break down every number in the list down to its prime factors. This signifies that every number in the list can be expressed as a product of prime numbers. Doing this for each number:- 12 is \(2^2 * 3^1\),- 18 is \(2^1 * 3^2\), and- 30 is \(2^1 * 3^1 * 5^1\)
03

Identifying Common Prime Factors

From the prime factors identified in Step 2, we need to find the prime factors common to all the numbers. In our list, the common prime factors are 2 and 3.
04

Determining the GCF

The GCF is determined by taking the smallest exponent of every common prime factor and then multiplying them together. Hence, for this list, the smallest exponent of 2 is 1 and for 3 is 1. Thus, GCF is \(2^1 * 3^1 = 6\)

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Most popular questions from this chapter

Use the \([\mathrm{GRAPH}]\) or \([\mathrm { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned}&3 x^{3}-12 x^{2}-15 x=3 x(x+5)(x-1) ;[-5,7,1] \text { by }\\\ &[-80,80,10] \end{aligned}$$

Use the \([\mathrm{GRAPH}]\) or \([\mathrm { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &6 x^{2}+10 x-4=2(3 x-1)(x+2) ;[-5,5,1] \text { by }\\\ &[-20,20,2] \end{aligned}$$

Describe a strategy that can be used to factor polynomials.

Use the \([\mathrm{GRAPH}]\) or \([\mathrm { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &x^{4}-16=\left(x^{2}+4\right)(x+2)(x-2) ;[-5,5,1] \text { by }\\\ &[-20,20,2] \end{aligned}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$3 x^{4}-9 x^{3} y+3 x^{2} y^{2}$$
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