# Find the greatest common factor (GCF/HCF) of the polynomial $9x^2, 15x^2y^3, 6xy^2$ and $21x^2y^2$.

Given:

Given polynomials are $9x^2, 15x^2y^3, 6xy^2$ and $21x^2y^2$.

To do:

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

Solution:

GCF/HCF:

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

The numerical coefficient of $9x^2$ is $9$

The numerical coefficient of $15x^2y^3$ is $15$

The numerical coefficient of $6xy^2$ is $6$

The numerical coefficient of $21x^2y^2$ is $21$

This implies,

$9=3\times3$

$15=3\times5$

$6=2\times3$

$21=3\times7$

HCF of $9, 15, 6$ and $21$ is $3$

The common variables in the given polynomials are $x$ and $y$

The power of $x$ in $9x^2$ is $2$

The power of $x$ in $15x^2y^3$ is $2$

The power of $x$ in $6xy^2$ is $1$

The power of $x$ in $21x^2y^2$ is $2$

The power of $y$ in $9x^2$ is $0$

The power of $y$ in $15x^2y^3$ is $3$

The power of $y$ in $6xy^2$ is $2$

The power of $y$ in $21x^2y^2$ is $2$

The monomial of common literals with the smallest power is $xy^0=x$

Therefore,

The greatest common factor of the given polynomials is $3x$.