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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

#### 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.

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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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