Question

How do you find the GCF of \[17x{{y}^{4}}+51{{x}^{3}}{{y}^{3}}\] ?

Answer 1

Hint: We are given an equation \[17x{{y}^{4}}+51{{x}^{3}}{{y}^{3}}\] and we have to find the GCF of the same. GCF refers to the Greatest Common Factor, that is, the term which is common in the terms involved. In the given question, we will simply take out the common components present in both the terms involved, which is, \[17x{{y}^{4}}\] and \[51{{x}^{3}}{{y}^{3}}\]. The common components from these terms will be the GCF.

Complete step by step solution:
According to the given question, we are given an equation \[17x{{y}^{4}}+51{{x}^{3}}{{y}^{3}}\] whose GCF we have to find out.
GCF refers to the Greatest Common Factor and is usually the common components from the terms involved or are under consideration.
For example – if we have 15 and 20, the GCF will be,
\[15=3\times 5\]
\[20=2\times 2\times 5\]
As we can see that 5 is common in both the numbers involved, so we can write that the GCF of the numbers 15 and 20 is,
\[GCF(15,20)=5\]
So, the equation that we have is,
\[17x{{y}^{4}}+51{{x}^{3}}{{y}^{3}}\]-----(1)
From the equation (1), we will take out the common terms and we get,
\[\Rightarrow 17x{{y}^{3}}(y+3{{x}^{2}})\]
Here, \[17x{{y}^{3}}\] is the term that is present in both the terms involved in GCF. And so, we have the final as,
\[GCF=17x{{y}^{3}}\]

Therefore, \[GCF=17x{{y}^{3}}\].

Note: The GCF we used in the above solution is different from the LCM. In LCM, we take all the distinct terms and if some terms are the same then we take only one of those. But, in GCF, we can have only those terms which are common in the terms involved in GCF.