What Is The Greatest Common Factor Of \[$-10x - 7\$\]?$\[ GCF = \square \\]

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Introduction

The greatest common factor (GCF) of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. In the context of algebra, we can also find the GCF of algebraic expressions. In this article, we will focus on finding the GCF of the given expression −10x−7-10x - 7.

Understanding the Concept of GCF

Before we dive into finding the GCF of −10x−7-10x - 7, let's briefly review the concept of GCF. The GCF of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

Factoring Out the GCF

To find the GCF of −10x−7-10x - 7, we need to factor out the greatest common factor. The GCF of −10x−7-10x - 7 is the largest number that divides both −10x-10x and −7-7 without leaving a remainder.

Finding the GCF

To find the GCF of −10x−7-10x - 7, we can start by listing the factors of −10x-10x and −7-7. The factors of −10x-10x are 1,−1,2,−2,5,−5,10,−10,x,−x,2x,−2x,5x,−5x,10x,−10x1, -1, 2, -2, 5, -5, 10, -10, x, -x, 2x, -2x, 5x, -5x, 10x, -10x. The factors of −7-7 are 1,−1,7,−71, -1, 7, -7.

Identifying the Greatest Common Factor

By comparing the factors of −10x-10x and −7-7, we can see that the greatest common factor is 11. The GCF of −10x−7-10x - 7 is 11.

Conclusion

In conclusion, the greatest common factor of −10x−7-10x - 7 is 11. This means that the largest number that divides both −10x-10x and −7-7 without leaving a remainder is 11.

Example Problems

Here are some example problems that involve finding the GCF of algebraic expressions:

  • Find the GCF of 6x+126x + 12 and 3x+63x + 6.
  • Find the GCF of 8x2+16x8x^2 + 16x and 4x2+8x4x^2 + 8x.
  • Find the GCF of 9x2+27x9x^2 + 27x and 3x2+9x3x^2 + 9x.

Step-by-Step Solutions

Here are the step-by-step solutions to the example problems:

Example Problem 1

Find the GCF of 6x+126x + 12 and 3x+63x + 6.

  • List the factors of 6x+126x + 12: 1,−1,2,−2,3,−3,6,−6,x,−x,2x,−2x,3x,−3x,6x,−6x1, -1, 2, -2, 3, -3, 6, -6, x, -x, 2x, -2x, 3x, -3x, 6x, -6x.
  • List the factors of 3x+63x + 6: 1,−1,3,−3,x,−x,3x,−3x1, -1, 3, -3, x, -x, 3x, -3x.
  • Identify the greatest common factor: The greatest common factor is 33.

Example Problem 2

Find the GCF of 8x2+16x8x^2 + 16x and 4x2+8x4x^2 + 8x.

  • List the factors of 8x2+16x8x^2 + 16x: 1,−1,2,−2,4,−4,8,−8,x,−x,2x,−2x,4x,−4x,8x,−8x,x2,−x2,2x2,−2x2,4x2,−4x2,8x2,−8x21, -1, 2, -2, 4, -4, 8, -8, x, -x, 2x, -2x, 4x, -4x, 8x, -8x, x^2, -x^2, 2x^2, -2x^2, 4x^2, -4x^2, 8x^2, -8x^2.
  • List the factors of 4x2+8x4x^2 + 8x: 1,−1,2,−2,4,−4,x,−x,2x,−2x,4x,−4x,x2,−x2,2x2,−2x2,4x2,−4x21, -1, 2, -2, 4, -4, x, -x, 2x, -2x, 4x, -4x, x^2, -x^2, 2x^2, -2x^2, 4x^2, -4x^2.
  • Identify the greatest common factor: The greatest common factor is 4x4x.

Example Problem 3

Find the GCF of 9x2+27x9x^2 + 27x and 3x2+9x3x^2 + 9x.

  • List the factors of 9x2+27x9x^2 + 27x: 1,−1,3,−3,9,−9,x,−x,3x,−3x,9x,−9x,x2,−x2,3x2,−3x2,9x2,−9x21, -1, 3, -3, 9, -9, x, -x, 3x, -3x, 9x, -9x, x^2, -x^2, 3x^2, -3x^2, 9x^2, -9x^2.
  • List the factors of 3x2+9x3x^2 + 9x: 1,−1,3,−3,x,−x,3x,−3x,x2,−x2,3x2,−3x21, -1, 3, -3, x, -x, 3x, -3x, x^2, -x^2, 3x^2, -3x^2.
  • Identify the greatest common factor: The greatest common factor is 3x3x.

Final Thoughts

In conclusion, finding the greatest common factor of algebraic expressions is an essential skill in mathematics. By following the steps outlined in this article, you can find the GCF of any algebraic expression. Remember to list the factors of each expression, identify the greatest common factor, and simplify the expression to find the final answer.

Frequently Asked Questions

Here are some frequently asked questions about finding the GCF of algebraic expressions:

  • What is the greatest common factor of −12x+18-12x + 18 and −6x+9-6x + 9?
  • What is the greatest common factor of 15x2+30x15x^2 + 30x and 5x2+10x5x^2 + 10x?
  • What is the greatest common factor of 24x2+48x24x^2 + 48x and 12x2+24x12x^2 + 24x?

Step-by-Step Solutions to Frequently Asked Questions

Here are the step-by-step solutions to the frequently asked questions:

Frequently Asked Question 1

Find the GCF of −12x+18-12x + 18 and −6x+9-6x + 9.

  • List the factors of −12x+18-12x + 18: 1,−1,2,−2,3,−3,6,−6,12,−12,x,−x,2x,−2x,3x,−3x,6x,−6x,12x,−12x1, -1, 2, -2, 3, -3, 6, -6, 12, -12, x, -x, 2x, -2x, 3x, -3x, 6x, -6x, 12x, -12x.
  • List the factors of −6x+9-6x + 9: 1,−1,3,−3,6,−6,x,−x,3x,−3x,6x,−6x1, -1, 3, -3, 6, -6, x, -x, 3x, -3x, 6x, -6x.
  • Identify the greatest common factor: The greatest common factor is 33.

Frequently Asked Question 2

Find the GCF of 15x2+30x15x^2 + 30x and 5x2+10x5x^2 + 10x.

  • List the factors of 15x2+30x15x^2 + 30x: 1,−1,3,−3,5,−5,15,−15,x,−x,3x,−3x,5x,−5x,15x,−15x,x2,−x2,3x2,−3x2,5x2,−5x2,15x2,−15x21, -1, 3, -3, 5, -5, 15, -15, x, -x, 3x, -3x, 5x, -5x, 15x, -15x, x^2, -x^2, 3x^2, -3x^2, 5x^2, -5x^2, 15x^2, -15x^2.
  • List the factors of 5x2+10x5x^2 + 10x: 1,−1,5,−5,x,−x,5x,−5x,x2,−x2,5x2,−5x21, -1, 5, -5, x, -x, 5x, -5x, x^2, -x^2, 5x^2, -5x^2.
  • Identify the greatest common factor: The greatest common factor is 5x5x.

Frequently Asked Question 3

Find the GCF of 24x2+48x24x^2 + 48x and 12x2+24x12x^2 + 24x.

  • List the factors of 24x2+48x24x^2 + 48x: $1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 24, -24, x, -x, 2x, -2x, 3x, -3x, 4x, -4x, 6x, -6x, 8x, -8x, 12x, -12x, 24x, -24x, x^2, -x^2, 2x^2, -2x^2, 3x^2, -3x^2, 4x^2, -4x^2, 6x^2, -6x^2, 8x^2, -8x^2, 12x^

Introduction

Finding the greatest common factor (GCF) of algebraic expressions is an essential skill in mathematics. In this article, we will answer some frequently asked questions about GCF and provide step-by-step solutions to help you understand the concept better.

Q1: What is the greatest common factor of −12x+18-12x + 18 and −6x+9-6x + 9?

A1

To find the GCF of −12x+18-12x + 18 and −6x+9-6x + 9, we need to list the factors of each expression and identify the greatest common factor.

  • List the factors of −12x+18-12x + 18: 1,−1,2,−2,3,−3,6,−6,12,−12,x,−x,2x,−2x,3x,−3x,6x,−6x,12x,−12x1, -1, 2, -2, 3, -3, 6, -6, 12, -12, x, -x, 2x, -2x, 3x, -3x, 6x, -6x, 12x, -12x.
  • List the factors of −6x+9-6x + 9: 1,−1,3,−3,6,−6,x,−x,3x,−3x,6x,−6x1, -1, 3, -3, 6, -6, x, -x, 3x, -3x, 6x, -6x.
  • Identify the greatest common factor: The greatest common factor is 33.

Q2: What is the greatest common factor of 15x2+30x15x^2 + 30x and 5x2+10x5x^2 + 10x?

A2

To find the GCF of 15x2+30x15x^2 + 30x and 5x2+10x5x^2 + 10x, we need to list the factors of each expression and identify the greatest common factor.

  • List the factors of 15x2+30x15x^2 + 30x: 1,−1,3,−3,5,−5,15,−15,x,−x,3x,−3x,5x,−5x,15x,−15x,x2,−x2,3x2,−3x2,5x2,−5x2,15x2,−15x21, -1, 3, -3, 5, -5, 15, -15, x, -x, 3x, -3x, 5x, -5x, 15x, -15x, x^2, -x^2, 3x^2, -3x^2, 5x^2, -5x^2, 15x^2, -15x^2.
  • List the factors of 5x2+10x5x^2 + 10x: 1,−1,5,−5,x,−x,5x,−5x,x2,−x2,5x2,−5x21, -1, 5, -5, x, -x, 5x, -5x, x^2, -x^2, 5x^2, -5x^2.
  • Identify the greatest common factor: The greatest common factor is 5x5x.

Q3: What is the greatest common factor of 24x2+48x24x^2 + 48x and 12x2+24x12x^2 + 24x?

A3

To find the GCF of 24x2+48x24x^2 + 48x and 12x2+24x12x^2 + 24x, we need to list the factors of each expression and identify the greatest common factor.

  • List the factors of 24x2+48x24x^2 + 48x: 1,−1,2,−2,3,−3,4,−4,6,−6,8,−8,12,−12,24,−24,x,−x,2x,−2x,3x,−3x,4x,−4x,6x,−6x,8x,−8x,12x,−12x,24x,−24x,x2,−x2,2x2,−2x2,3x2,−3x2,4x2,−4x2,6x2,−6x2,8x2,−8x2,12x2,−12x2,24x2,−24x21, -1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 24, -24, x, -x, 2x, -2x, 3x, -3x, 4x, -4x, 6x, -6x, 8x, -8x, 12x, -12x, 24x, -24x, x^2, -x^2, 2x^2, -2x^2, 3x^2, -3x^2, 4x^2, -4x^2, 6x^2, -6x^2, 8x^2, -8x^2, 12x^2, -12x^2, 24x^2, -24x^2.
  • List the factors of 12x2+24x12x^2 + 24x: 1,−1,2,−2,3,−3,4,−4,6,−6,12,−12,x,−x,2x,−2x,3x,−3x,4x,−4x,6x,−6x,12x,−12x,x2,−x2,2x2,−2x2,3x2,−3x2,4x2,−4x2,6x2,−6x2,12x2,−12x21, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12, x, -x, 2x, -2x, 3x, -3x, 4x, -4x, 6x, -6x, 12x, -12x, x^2, -x^2, 2x^2, -2x^2, 3x^2, -3x^2, 4x^2, -4x^2, 6x^2, -6x^2, 12x^2, -12x^2.
  • Identify the greatest common factor: The greatest common factor is 12x12x.

Q4: How do I find the greatest common factor of two or more numbers?

A4

To find the greatest common factor of two or more numbers, you need to list the factors of each number and identify the greatest common factor. Here's a step-by-step guide:

  1. List the factors of each number.
  2. Identify the common factors among the numbers.
  3. Determine the greatest common factor among the common factors.

Q5: What is the greatest common factor of −18x+24-18x + 24 and −6x+8-6x + 8?

A5

To find the GCF of −18x+24-18x + 24 and −6x+8-6x + 8, we need to list the factors of each expression and identify the greatest common factor.

  • List the factors of −18x+24-18x + 24: 1,−1,2,−2,3,−3,6,−6,9,−9,18,−18,x,−x,2x,−2x,3x,−3x,6x,−6x,9x,−9x,18x,−18x1, -1, 2, -2, 3, -3, 6, -6, 9, -9, 18, -18, x, -x, 2x, -2x, 3x, -3x, 6x, -6x, 9x, -9x, 18x, -18x.
  • List the factors of −6x+8-6x + 8: 1,−1,2,−2,4,−4,6,−6,x,−x,2x,−2x,4x,−4x,6x,−6x1, -1, 2, -2, 4, -4, 6, -6, x, -x, 2x, -2x, 4x, -4x, 6x, -6x.
  • Identify the greatest common factor: The greatest common factor is 22.

Q6: What is the greatest common factor of −20x2+40x-20x^2 + 40x and −10x2+20x-10x^2 + 20x?

A6

To find the GCF of −20x2+40x-20x^2 + 40x and −10x2+20x-10x^2 + 20x, we need to list the factors of each expression and identify the greatest common factor.

  • List the factors of −20x2+40x-20x^2 + 40x: 1,−1,2,−2,4,−4,5,−5,10,−10,20,−20,x,−x,2x,−2x,4x,−4x,5x,−5x,10x,−10x,20x,−20x,x2,−x2,2x2,−2x2,4x2,−4x2,5x2,−5x2,10x2,−10x2,20x2,−20x21, -1, 2, -2, 4, -4, 5, -5, 10, -10, 20, -20, x, -x, 2x, -2x, 4x, -4x, 5x, -5x, 10x, -10x, 20x, -20x, x^2, -x^2, 2x^2, -2x^2, 4x^2, -4x^2, 5x^2, -5x^2, 10x^2, -10x^2, 20x^2, -20x^2.
  • List the factors of −10x2+20x-10x^2 + 20x: 1,−1,2,−2,5,−5,10,−10,x,−x,2x,−2x,5x,−5x,10x,−10x,x2,−x2,2x2,−2x2,5x2,−5x2,10x2,−10x21, -1, 2, -2, 5, -5, 10, -10, x, -x, 2x, -2x, 5x, -5x, 10x, -10x, x^2, -x^2, 2x^2, -2x^2, 5x^2, -5x^2, 10x^2, -10x^2.
  • Identify the greatest common factor: The greatest common factor is 10x10x.

Q7: What is the greatest common factor of −30x2+60x-30x^2 + 60x and −15x2+30x-15x^2 + 30x?

A7

To find the GCF of −30x2+60x-30x^2 + 60x and −15x2+30x-15x^2 + 30x, we need to list the factors of each expression and identify the greatest common factor.

  • List the factors of −30x2+60x-30x^2 + 60x: $1, -1, 2, -2, 3, -3, 5, -5, 6, -6, 10, -10, 15, -15, 30, -30, x, -x, 2x, -2x, 3x, -3x, 5x, -5x, 6x, -6x, 10x, -10x, 15x, -15x, 30x, -30x, x^2, -x^2, 2x^2, -2x^2, 3x^2, -3x^2, 5x^2, -5x^2, 6x^2, -6x^2, 10