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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 $3x$.
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