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Chapter 7: Problem 18

Find the greatest common factor. \(27 p^{2}, 45 p^{3}, 9 p^{4}\)

Short Answer

Expert verified
The greatest common factor is 9p^{2}.

Step by step solution

01

Identify the coefficients

The coefficients of the terms are 27, 45, and 9. The aim is to find the greatest common factor (GCF) of these numbers.
02

Find the GCF of the coefficients

The factors of 27 are 1, 3, 9, and 27. The factors of 45 are 1, 3, 5, 9, 15, and 45. The factors of 9 are 1, 3, and 9. The greatest common factor of these coefficients is 9.
03

Identify the variables

The terms contain the variable p with powers 2, 3, and 4. The greatest common factor of these powers is the smallest power, which is 2.
04

Combine the GCF of coefficients and variables

Putting together the GCF of the coefficients (9) and the GCF of the variable's lowest power ( p^{2} ), the GCF of the terms is 9p^{2}.

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Key Concepts

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

Understanding Coefficients
Coefficients are the numerical parts of terms in algebra. For the terms given, 27, 45, and 9 are the coefficients.
It鈥檚 important to find the greatest common factor (GCF) of these numbers to simplify expressions.
The GCF is the largest number that divides all the given numbers equally.
Finding the factors of each coefficient helps us identify the GCF:
  • Factors of 27: 1, 3, 9, 27
  • Factors of 45: 1, 3, 5, 9, 15, 45
  • Factors of 9: 1, 3, 9
The greatest common factor among them is 9.
Understanding coefficients helps us break down and simplify algebraic expressions.
Demystifying Variables
Variables are letters or symbols that represent unknown numbers or quantities in algebra.
In the problem, the variable is denoted as\(p\) and it has different powers: 2, 3, and 4.
When finding the greatest common factor involving variables, focus on the smallest power for commonality.
In this scenario, the smallest power is 2. Hence, the variable part of the GCF is\(p^2\).
This means any common factor of these terms must include\(p^2\).
Understanding variables and their powers is crucial for working with algebraic expressions effectively.
Factoring in Algebra
Factoring in algebra involves breaking down expressions into simpler components that can be multiplied together to get the original expression.
In this exercise, we factor the terms by finding their GCF:
First, we determined the GCF of the coefficients, which is 9. Then, we took the lowest power of the variable
\(p^{2}\).
By combining these, we get the GCF of the entire expression: 9p^{2}.
Factoring helps simplify problems and makes it easier to solve equations or further manipulate expressions.
This is a key skill in algebra, allowing us to work with complex expressions in a more manageable form.

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