What is the Greatest Common Factor of 42a5b3, 35a3b4, and 42ab4? A Step-by-Step Guide

Understanding how to find the greatest common factor (GCF) of algebraic expressions is crucial for solving many mathematical problems. This article explains what is the greatest common factor of 42a5b3, 35a3b4, and 42ab4, with a step-by-step breakdown of the process. The GCF is the largest expression that divides each of the given expressions without leaving a remainder. Let’s delve deeper to find the answer.

1. Breaking Down the Algebraic Expressions

To find out what is the greatest common factor of 42a5b3, 35a3b4, and 42ab4, we first need to analyze each expression:

42a5b3: This expression consists of the numerical coefficient 42, and the variables $a$ and $b$ raised to the powers 5 and 3, respectively.
35a3b4: This expression has a numerical coefficient of 35, with variables $a$ and $b$ raised to the powers 3 and 4, respectively.
42ab4: This expression includes the numerical coefficient 42, with variable $a$ raised to the power 1 and $b$ raised to the power 4.

Each expression is composed of a coefficient (numerical part) and variables with their respective exponents (algebraic part). We need to separately find the GCF of the coefficients and the GCF of the variables.

2. Finding the GCF of the Numerical Coefficients

To determine what is the greatest common factor of 42a5b3, 35a3b4, and 42ab4, we start by finding the GCF of the numerical coefficients: 42, 35, and 42.

Prime factorization of 42: $\times 3 \times 7$
Prime factorization of 35: $\times 7$
Prime factorization of 42 (again): $\times 3 \times 7$

The common prime factor in all three coefficients is $7$ . Therefore, the GCF of the numerical coefficients is 7.

3. Determining the GCF of the Variable Parts

To completely understand what is the greatest common factor of 42a5b3, 35a3b4, and 42ab4, we also need to find the GCF of the variable parts for $a$ and $b$ .

Step 1: Finding the GCF for Variable $a$

In $42 a 5 b 3$ , the power of $a$ is 5.
In $35 a 3 b 4$ , the power of $a$ is 3.
In $42 ab 4$ , the power of $a$ is 1.

To find the GCF for variable $a$ , we take the smallest power, which is $a^1$ .

Step 2: Finding the GCF for Variable $b$

In $42 a 5 b 3$ , the power of $b$ is 3.
In $35 a 3 b 4$ , the power of $b$ is 4.
In $42 ab 4$ , the power of $b$ is 4.

For variable $b$ , the smallest power is $b^3$ .

4. Combining the Results to Find the GCF

After determining the GCF of both the numerical coefficients and the variable parts, we combine them to find out what is the greatest common factor of 42a5b3, 35a3b4, and 42ab4:

$GCF=7×a1×b3=7ab3\text{GCF} = 7 \times a^1 \times b^3 = 7ab^3$

Hence, the Greatest Common Factor of the expressions $42 a 5 b 3$ , $35 a 3 b 4$ , and $42 ab 4$ is $7ab^3$ .

5. Validating the GCF

To confirm that $7ab^3$ is indeed the correct answer to the question what is the greatest common factor of 42a5b3, 35a3b4, and 42ab4, let’s divide each expression by $7ab^3$ :

For $42 a 5 b 3$ :
$42a5b37ab3=6a4\frac{42a^5b^3}{7ab^3} = 6a^4$ This division yields a simplified expression, verifying that $7ab^3$ divides $42 a 5 b 3$ .
For $35 a 3 b 4$ :
$35a3b47ab3=5a2b\frac{35a^3b^4}{7ab^3} = 5a^2b$ This also results in a simplified expression, confirming that $7ab^3$ divides $35 a 3 b 4$ .
For $42 ab 4$ :
$42ab47ab3=6b\frac{42ab^4}{7ab^3} = 6b$ The result is a valid expression, confirming that $7ab^3$ divides $42ab^4$ .

Since $7ab^3$ divides all three expressions without leaving a remainder, this confirms our answer.

6. Importance of Finding the Greatest Common Factor

Understanding what is the greatest common factor of 42a5b3, 35a3b4, and 42ab4 helps in simplifying algebraic expressions, solving equations, and factoring polynomials. The GCF is a fundamental concept that has practical applications, including:

Simplifying Fractions: Simplifying algebraic fractions by dividing both the numerator and the denominator by their GCF.
Factoring Polynomials: In polynomial equations, the initial step often involves factoring out the GCF to simplify the expression into smaller factors.
Solving Equations: Algebraic equations can be simplified by factoring out the GCF, making them easier to solve.

7. Conclusion

Now that we have thoroughly explored what is the greatest common factor of 42a5b3, 35a3b4, and 42ab4, we know that the GCF is $7ab^3$ . This result is achieved by breaking down the coefficients and variables separately and combining their lowest common factors. Determining the GCF is a crucial skill in algebra, number theory, and many areas involving mathematical simplification and optimization.

By mastering how to find the GCF, one can simplify complex problems, making them easier to manage and solve. This skill is fundamental not only in mathematics but also in fields like engineering, computer science, and cryptography, where effective problem-solving and optimization are essential. Read More D2armorpicker.