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  4. If the polynomials f(x)=x2+21 x+m and g(x)=x2+22 x+n have a common factor, then find the value of (m-n)2

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Q. If the polynomials $f(x)=x^2+21 x+m$ and $g(x)=x^2+22 x+n$ have a common factor, then find the value of $(m-n)^2$ ___

Remainder and Factor Theorems

Solution:

Let the common factor be $x-a$ for both $f(x)$ and $g(x)$
Now, $x-a$ is a factor
So, $f(a)=a^2+21 a+m=0$
$ a^2+21 a+m=0 $
$ a^2=-21 a-m$...(i)
Also, $g(a)=a^2+22 a+n=0$
$ a^2+22 a+n=0 $
$ a^2=-22 a-n$...(ii)
From (i) and (ii),
$ -21 a-m=-22 a-n $
$ \Rightarrow m-n=a \Rightarrow(m-n)^2=a^2 $
$ \Rightarrow(m-n)^2=-21 a-m \text { or }(m-n)^2=-22 a-n$
$ \Rightarrow(m-n)^2=-21(m-n)-m \text { or }$
$ (m-n)^2=-22(m-n)-n $
$ \Rightarrow(m-n)^2=-21 m+21 n-m \text { or }$
$ (m-n)^2=-22 m+22 n-n$
$ \Rightarrow(m-n)^2=-22 m+21 n \text { or } $
$ (m-n)^2=-22 m+21 n$