# Find the greatest common factor (GCF/HCF) of the polynomials $36a^2b^2c^4, 54a^5c^2$ and $90a^4b^2c^2$.

Given:

Given polynomials are $36a^2b^2c^4, 54a^5c^2$ and $90a^4b^2c^2$.

To do:

We have to find the greatest common factor of the given polynomials.

Solution:

HCF:

A common factor of two or more numbers is a factor that is shared by the numbers. The highest common factor (HCF) of those numbers is found by finding all common factors of the numbers and selecting the largest one.

The numerical coefficient of $36a^2b^2c^4$ is $36$

The numerical coefficient of $54a^5c^2$ is $54$

The numerical coefficient of $90a^4b^2c^2$ is $90$

This implies,

$36=2\times2\times3\times3$

$54=2\times3\times3\times3$

$90=2\times3\times3\times5$

HCF of $36, 54$ and $90$ is $2\times3\times3=18$

The common variables in the given polynomials are $a, b$ and $c$

The power of $a$ in $36a^2b^2c^4$ is $2$

The power of $a$ in $54a^5c^2$ is $5$

The power of $a$ in $90a^4b^2c^2$ is $4$

The power of $b$ in $36a^2b^2c^4$ is $2$

The power of $b$ in $54a^5c^2$ is $0$

The power of $b$ in $90a^4b^2c^2$ is $2$

The power of $c$ in $36a^2b^2c^4$ is $4$

The power of $c$ in $54a^5c^2$ is $2$

The power of $c$ in $90a^4b^2c^2$ is $2$

The monomial of common literals with the smallest power is $a^2b^0c^2=a^2c^2$

Therefore,

The greatest common factor of the given polynomials is $18a^2c^2$.