Question

If the HCF of polynomials $ f(x) $ and $ g(x) $ is \[4x - 6\] , then $ f(x) $ and $ g(x) $ could be:
a) $ 2,2x - 3 $
b)\[8x - 12,2\]
c)\[2{(2x - 3)^2},4(2x - 3)\]
d)\[2\left( {2x - 3} \right),4\left( {2x + 3} \right)\]

Answer 1

Hint: HCF of two polynomial needs to be a common factor of both the polynomials .so we will try to factories the given polynomials to the simplest form then we will try to figure out from the given solution which of the above will have a common factor of \[4x - 6\] or its factories value.

Complete step-by-step answer:
Since HCF stands for highest common factor we will first multiply the given polynomials and try to factories the given polynomial and find out the simplest form if it exists.
 So, we are given in the question that HCF of two polynomials is \[4x - 6\]
Let’s go through these given options
1. First, we have $ 2,2x - 3 $ ,
 Now $ 2 $ as the first polynomial which don’t have a factor for \[4x - 6\]
So the first option is discarded.
Hence, option a is not the correction option.
2. Now we have second option as \[8x - 12,2\]
We have \[2\] as the first polynomial which don’t have a factor for \[4x - 6\]
Hence option b is also discarded.
3. Now we have \[2{(2x - 3)^2},4(2x - 3)\]
Clearly, both the given polynomials have \[2(2x - 3)\]
And \[2(2x - 3) = 4x - 6\] which is the required factor.
Hence this option is correct.
  Then we have 2(2x-3)
Now another term is 2x-3
 $ \Rightarrow $ hence the common factor is 2x-3
Hence option is also not the correct answer.
4. Now we are left with \[2\left( {2x - 3} \right),4\left( {2x + 3} \right)\]
Clearly, we have common factor $ 2 $
Hence, the option (d) is also discarded.
Therefore, among all the four choices option (c) is correct.
So, the correct answer is “Option C”.

Note: After solving for first three options we could get that the first two are not the correct option and the third one is the correct option. Luckily in our case the last option turned out to be an incorrect option, but for clarification always try to check all the four options.