Question

For Exercises, find the GCF of the monomials. \(42 x^{4}, 35 x^{6},\) and \(28 x^{3}\)

Short answer

Answer: The GCF of the given monomials is \(7x^{3}\).

Step-by-step solution

Identify the coefficients and powers of x

The monomials are \(42x^{4}, 35x^{6}\), and \(28x^{3}\). The coefficients are 42, 35, and 28. The powers of x are 4, 6, and 3.

Find the GCF of the coefficients

To find the GCF of the coefficients 42, 35, and 28, we can list their factors and find the greatest number that is common among these factors. The factors are: - Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 - Factors of 35: 1, 5, 7, 35 - Factors of 28: 1, 2, 4, 7, 14, 28 The greatest common factor of these coefficients is 7.

Find the GCF of the powers of x

We have powers of x as 4, 6, and 3. The GCF of these powers is the smallest power among them, which is 3.

Combine the GCF of coefficients and powers of x

We found the GCF of the coefficients to be 7 and the GCF of the powers of x to be 3. So, the GCF of the given monomials is \(7x^{3}\).

Key concepts

Understanding Monomials

Monomials are algebraic expressions that consist of a single term. They are simply a combination of numbers and variables, often represented in the form of \( ax^n \), where \( a \) is a coefficient and \( n \) is a non-negative integer indicating the power of the variable.
For example, in this exercise, we have monomials like \(42x^4, 35x^6,\) and \(28x^3\). Each of these monomials includes a coefficient and a power of \( x \).
Monomials can involve:

• Pure constants (like 42, 35, and 28)
• Variables with positive integer exponents (like \(x^4, x^6, x^3\))

Monomials play a vital role in understanding algebraic expressions since they form the building blocks of more complex polynomial expressions.

Factors in Mathematics

Factors are numbers or expressions that can be multiplied to produce another number or expression. In the context of our exercise, factors are crucial for determining the Greatest Common Factor (GCF) of given coefficients.
To find the GCF of numbers like 42, 35, and 28, we need to:

• List all factors for each number
• Identify the largest number that appears in each list

For instance, after finding the factors:
- Factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42
- Factors of 35 are 1, 5, 7, 35
- Factors of 28 are 1, 2, 4, 7, 14, 28
The greatest common factor here is 7. This means 7 is the largest number that divides each of these numbers without leaving a remainder.

Identifying Coefficients

Coefficients are the numerical part of terms in an algebraic expression. They are the numbers placed in front of variables. In our provided monomials, the coefficients are 42, 35, and 28.
Coefficients have a direct impact on the values produced by an expression, as they scale the effect of the accompanying variable's powers.
When finding the GCF of monomials, it is essential to focus first on the coefficients. Always find the GCF of the coefficients separately to simplify the process.
In algebra, the role of coefficients extends beyond mere numbers:

• They signify how much of the variable's effect is present.
• They affect the overall size and direction (positive or negative) of the term.

Understanding coefficients helps make calculations of the GCF more manageable and aligns with procedures used in the GCF of monomials.