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Chapter 6: Problem 8

Find the greatest common factor of each list of monomials. $$10 y^{5}, 20 y^{2},\( and \)-5 y$$

Short Answer

Expert verified
The greatest common factor of the monomials \( 10 y^{5}, 20 y^{2}, \) and \( -5 y \) is \( 5y \).

Step by step solution

01

Determine the GCF of the numerical coefficients

First, find the GCF of the numerical coefficients 10, 20, and -5. The GCF of these numbers is 5.
02

Determine the GCF of the variables

Next, find the GCF of the variables. Here, the given monomials have y with different powers, namely 5, 2, and 1. The GCF is the variable with the lowest power. So, the GCF of these variables is \( y^1 \) or simply y.
03

Write the GCF of the monomials

Finally, write the GCF by combining the GCF of the numerical coefficients and the variables. Hence, the GCF of the given monomials \( 10 y^{5}, 20 y^{2}, \) and \( -5 y \) is \( 5y \).

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Key Concepts

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

Monomials
Monomials are algebraic expressions that consist of a single term. This means they can be as simple as a number, a variable, or a product of numbers and variables. Monomials do not include addition or subtraction; they are purely multiplicative. For example, in the expression you鈥檙e working with, each term like \(10y^5\) or \(-5y\) is called a monomial because it doesn't have any addition or subtraction parts.
These expressions can include:
  • Numerical coefficients: These are numbers in front of the variables. In \(10y^5\), 10 is the coefficient.
  • Variables: These letters represent numbers. For example, \(y\) can represent any number.
  • Exponents (optional): Exponents indicate how many times to multiply the variable by itself. In \(y^5\), 5 is the exponent.
This simple structure makes it easier to find the greatest common factor of monomials!
Numerical Coefficients
Numerical coefficients are numbers located in front of variables in an algebraic expression or monomial. They tell us how many times the variable should be counted. For example, in the monomials given - \(10y^5\), \(20y^2\), and \(-5y\) - the numbers 10, 20, and -5 are the numerical coefficients.
To find the greatest common factor (GCF) of these numerical coefficients, you should:
  • Identify each coefficient: 10, 20, and -5.
  • List the factors of each number:
    • Factors of 10 are 1, 2, 5, 10.
    • Factors of 20 are 1, 2, 4, 5, 10, 20.
    • Factors of -5 are 1, 5, -1, -5.
  • Find the highest common factor: The common factors are 1 and 5, so the GCF is 5.
Recognizing the GCF among numerical coefficients simplifies combining these monomials in your calculations.
Variables
Variables, such as \(y\), are symbols representing numbers that we might not yet know. They are a core part of algebra and help make expressions dynamic and flexible. In your given monomials 鈥� \(10y^5\), \(20y^2\), and \(-5y\) 鈥� the letter \(y\) is the variable, and it's associated with different powers.
When finding the GCF for variables, the smallest exponent among the given terms is what you're looking for:
  • The powers of \(y\) are 5, 2, and 1.
  • Choose the smallest power as it can divide all given powers equally.
Therefore, the GCF for the variable part across these monomials is \(y^1\), or simply \(y\).
This concept ensures that when simplifying expressions, you only use shared elements, simplifying the calculations further.

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Most popular questions from this chapter

Exercises \(150-152\) will help you prepare for the material covered in the next section. Evaluate \((3 x-1)(x+2)\) for \(x=\frac{1}{3}\)

Solve equation and check your solutions. \(y^{3}+3 y^{2}+2 y=0\)

The length of a rectangular garden is 5 feet greater than the width. The area of the rectangle is 300 square feet. Find the length and the width.

Write a quadratic equation in standard form whose solutions are \(-3\) and 5

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \((x+1)(2 x+5)=-1\)
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