In this lesson, we are going to learn how to find the Greatest Common Factor (GCF) of the algebraic expressions: 16s³t, 40s⁵, and 68t². The GCF is the largest factor that divides all given terms without leaving a remainder.
Let’s go through the solution step by step, using a simple and straightforward approach.
Step 1: Break down each term into its prime factors
Start by factoring each coefficient (the numbers in front of the variables) and the variables themselves.
a. 16s³t:
Coefficient (16): The prime factorization of 16 is: 16=24
Variables: For s³t, we have: s3=s×s×s,t = t
So, 16s³t = 24×s3×t
b. 40s⁵:
Coefficient (40): The prime factorization of 40 is: 40=23×5
Variables: For s⁵, we have: s5=s×s×s×s×s
So, 40s⁵ = 23×5×s5
c. 68t²:
Coefficient (68): The prime factorization of 68 is: 68=22×17
Variables: For t², we have: t2=t×t
So, 68t² = 22×17×t2
Step 2: Identify the common factors
Now, we need to find the common factors from each term.
Coefficients: Let’s look at the numbers first.
- In 16, the factor is 24
- In 40, the factor is 23×5.
- In 68, the factor is 22×17
Variables:
s-term: The powers of s in the expressions are:
- 16s³t: s3s³s3
- 40s⁵: s5s⁵s5
- 68t²: No s here.
t-term: The powers of t are:
- 16s³t: t1t¹t1
- 40s⁵: No t here.
- 68t²: t2t²t2
Step 3: Multiply the common factors
Now that we have identified the common factors, we can multiply them to find the GCF.
- Coefficient common factor: 22=4
- Variable common factor: t1=t
So, the GCF of 16s³t, 40s⁵, and 68t² is: 4t
Final Answer: GCF = 4t
Recap of the steps:
- Factor the coefficients of each term into primes.
- Identify the lowest powers of the common factors for both the numbers and the variables.
- Multiply the common factors to find the GCF.
Thus, the GCF of 16s³t, 40s⁵, and 68t² is 4t. This means 4t is the largest factor that divides all three expressions without a remainder.
Practice Problem
Now, try finding the GCF of the following expressions:
- 18a²b, 24a⁴, and 30ab².
Use the same steps to break down each term and find the greatest common factor. Good luck and
If you feel difficulty finding the GCF then ask in the comment section or use our Online Free GCF Calculator