Question

Find the greatest common factor (GCF/HCF) of the following polynomial.
$42{x^2}yz$ and $63{x^3}{y^2}{z^3}$

Answer 1

Hint: Greatest common factor is also known as the highest common factor. It is the largest or the greatest common factor among the given two or more terms. To find the HCF we will find out its prime factors to simply find the common factors, here we will first find the GCF for the numeric terms and then for variables.

Complete step by step solution:
Here, in the given terms $42{x^2}yz$ and $63{x^3}{y^2}{z^3}$
The numerical coefficients are of the given numerical are $42$ and $63$
We will find the prime factors for both the terms. Factors are the terms which when multiplied gives the original value as the resultant term.
$
  42 = 2 \times 3 \times 7 \\
  63 = 3 \times 3 \times 7 \;
 $
From the above two expressions, we can observe the greatest common factor are $3 \times 7 = 21$ …(A)
Now, the common variables appeared with the constants are x, y and z
The smallest power of “x” in both the monomials is $ = 2$
The smallest power of “y” in both the monomials is $ = 1$
The smallest power of “z” in both the monomials is $ = 1$
So, all together the monomials of the common variables with the smallest power $ = {x^2}yz$ ….. (B)
From the equation (A) and (B)
Hence, the greatest common factor is $ = 21{x^2}yz$
So, the correct answer is “$ = 21{x^2}yz$”.

Note: Be always clear with the HCF and LCM (least common multiple) to solve these types of examples. HCF can be defined as the highest or greatest common multiple whereas the LCM is defined as the least common multiple or least common divisor in two or more given numbers.