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If X + a is a Common Factor of Expressions F(X) = X2 + Px + Q and G(X) = X2 + Mx + N; Show that : `A=(N-q)/(M-p)` - ICSE Class 10 - Mathematics

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Question

If x + a is a common factor of expressions f(x) = x2 + px + q and g(x) = x2 + mx + n; 

show that : `a=(n-q)/(m-p)` 

 

Solution

`f(x)=x^2+px+q` 

it is given that (x+a) is a factor of f(x) 

∴ f(-a)=0 

 ⇒`(-a)^2+p(-a)+q=0 ` 

 ⇒ `a^2-pa+q=0 `

 ⇒ `a^2=pa-q ` ........(1) 

`g(x)=x^2+mx+n `

it is given thhat (x+a) is a factor of g (x). 

∴ `g(-a)=0 `

  ⇒ `(-a)^2+m(-a)+n=0 `

  ⇒ `a^2-ma+n=0 `

  ⇒ `a^2=ma-n` ........(2) 

from (1) and (2), we get, 

`pa-q=ma-n `

`n-q=a(m-p) `

`a=(n-q)/(m-p)` 

Hence, proved.

  Is there an error in this question or solution?

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Solution If X + a is a Common Factor of Expressions F(X) = X2 + Px + Q and G(X) = X2 + Mx + N; Show that : `A=(N-q)/(M-p)` Concept: Factor Theorem.
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