Question

How do you find the greatest common factor of \[45{{y}^{12}}+30{{y}^{10}}\]?

Answer 1

Hint: The greatest common factor is the factor that divides a given number of terms. In order to find the greatest common factor of this polynomial, we write it in factors form of the prime numbers and then find the factor that is common in both the terms.

Formula used:
In order to find the prime factors, we start by dividing the number by the least prime number \[2\] and divide until we get a remainder. After that, we divide it by \[3,5,7\] and so on until we are left with prime numbers only .Secondly, we write all the prime numbers in the multiplication form.
For example for \[20\], we start by dividing \[20\] by\[2\] , and after dividing twice, we get \[5\] as remainder which is also a prime number so we write, \[\left( 2\times 2\times 5 \right)\].

Complete step by step solution:
In order to find the greatest common factor of the equation \[45{{y}^{12}}+30{{y}^{10}}\] , we need to follow the following steps:
Firstly, we need to write the prime factor form of the two terms of the polynomial equation:
 \[\begin{align}
  & 45{{y}^{12}}+30{{y}^{10}} \\
 & \Rightarrow \left( 3\times 3\times 5\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}} \right)+\left( 2\times 3\times 5\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}} \right) \\
\end{align}\]
Now, we find the factors that occur in both the terms of the equation and highlight them
\[\Rightarrow \left( 3\times 3\times 5\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}} \right)+\left( 2\times 3\times 5\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}} \right)\]
Take out the factors that are common in both the terms of the equation.
\[\begin{align}
  & \Rightarrow \left[ 3\times 5\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}}\times {{y}^{2}} \right]\left[ \left( 3\times {{y}^{2}} \right)+2 \right] \\
 & \Rightarrow 15{{y}^{10}}\left( 3{{y}^{2}}+2 \right) \\
\end{align}\]

Therefore, the greatest common factor for \[45{{y}^{12}}+30{{y}^{10}}\] is\[15{{y}^{10}}\].

Note: In this question, always start with the lowest prime factor only. Starting with a factor other than the lowest one may be troublesome. Writing \[\left( 3\times 15 \right)=45\]is factorization and not the prime factorization.