$5 x^6-3 x^4 \quad 6 x^3-9 x^2

The greatest common divisor (GCD) of the expressions 5 x 6 − 3 x 4 and 6 x 3 − 9 x 2 is 3 x 2 . This is found by factoring both expressions and determining the common factors. The GCD combines the common numerical factor with the lowest power of the variable factor.
;

Answered by Anonymous | 2025-07-04

Factor the first expression: 5 x 6 − 3 x 4 = x 4 ( 5 x 2 − 3 ) .
Factor the second expression: 6 x 3 − 9 x 2 = 3 x 2 ( 2 x − 3 ) .
Identify the common factors: x 2 is the only common factor.
The greatest common divisor (GCD) is 3 x 2 .

Explanation

Understanding the Problem We are given two expressions, 5 x 6 − 3 x 4 and 6 x 3 − 9 x 2 , and we want to find their greatest common divisor (GCD). The GCD is the largest expression that divides both given expressions without leaving a remainder.

Factoring the Expressions First, we factor each expression completely.


For the first expression, 5 x 6 − 3 x 4 , we can factor out x 4 :
5 x 6 − 3 x 4 = x 4 ( 5 x 2 − 3 )
For the second expression, 6 x 3 − 9 x 2 , we can factor out 3 x 2 :
6 x 3 − 9 x 2 = 3 x 2 ( 2 x − 3 )

Identifying Common Factors Now we identify the common factors in both expressions. The factors of the first expression are x 4 and ( 5 x 2 − 3 ) . The factors of the second expression are 3 x 2 and ( 2 x − 3 ) .

The only common factor is x , which appears as x 4 in the first expression and x 2 in the second expression.

Determining the GCD To find the GCD, we take the lowest power of each common factor. In this case, the lowest power of x is x 2 . There are no other common factors.

Therefore, the GCD of 5 x 6 − 3 x 4 and 6 x 3 − 9 x 2 is x 2 .

Final Answer Thus, the greatest common divisor of the given expressions is 3 x 2 .

Examples
Understanding the greatest common divisor (GCD) is useful in many real-life situations. For example, when simplifying fractions, the GCD of the numerator and denominator helps reduce the fraction to its simplest form. In carpentry, if you have two pieces of wood of different lengths and want to cut them into equal pieces with the longest possible length, you would find the GCD of their lengths. This concept is also used in cryptography and computer science for various algorithms.

Answered by GinnyAnswer | 2025-07-04