Solving the Greatest Common Factor of 4k, 18k4, and 12

Mathematics can be a challenging subject to deal with. Nevertheless, it is an essential subject that opens up many possibilities in the world of science and technology. Today, we will be discussing how to solve the greatest common factor (GCF) of 4k, 18k4, and 12.

Before we jump into the solution, let us first understand what a GCF is. The GCF of two or more numbers refers to the largest number that divides them without any remainder. In simpler terms, it is the largest number that is a factor of two or more numbers.

Now, let us look at the numbers we have been given – 4k, 18k4, and 12. We notice that all these numbers are multiples of 4. Therefore, we can start by factoring out the GCF of 4:

4(k + 4k3 + 3)

We can now see that we have 4, and the expression within the brackets contains k terms. If we take a look at the second term, we can notice that it contains k raised to the power of 3. In other words, it is a cubic polynomial.

Next, we have to determine if there are any other common factors between the remaining terms. To do this, we need to factor them out as well. Let us start with 18k4:

18k4 = 2 x 9 x k4

= 2 x 32 x k4

= 2 x 32 x (k2)2

We have now factored 18k4 into its prime factors, and we can see that it contains 2 and 3 raised to the power of 2. It also has k raised to the power of 4.

Let us now factor out the GCF from 2(k + 4k3 + 3):

2(k + 4k3 + 3) = 2 x (k + 4k3 + 3)

We can see that this expression also contains 2, which is a common factor with 18k4. However, we do not have to factor it out again since we have already done so.

Now, let us move on to factorizing the number 12:

12 = 2 x 2 x 3

= 22 x 3

We can see that 12 contains 2 raised to the power of 2 and 3. We can now factor out the GCF from the expression we obtained earlier:

2(k + 4k3 + 3)

= 2 x (k + 4k3 + 3)

The final step is to determine the GCF of all three expressions. We can see that the GCF is 2. Therefore, the greatest common factor of 4k, 18k4, and 12 is 2 x (k + 4k3 + 3).

In conclusion, finding the GCF of numbers can be solved by factoring out the common factors and determining the largest factor. At times, the process may seem challenging, especially when dealing with complicated expressions, but with continuous practice, it becomes more comfortable. The key takeaway from this exercise is to take one step at a time, analyze every expression methodically, and remember that the answer lies in the prime factors that make up the given numbers.