Question

If a and b are unequal and x^2 + ax + b and x^2 + bx + a have a common factor, then show that a + b + 1 = 0.

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Answer 1

#hope it helps u

[/tex]Let

P(x)=x2+ax+b

Q(x)=x2+bx+a

Given that P(x) and Q(x) have a common factor.

Let (x−α) be the common factor.

Therefore,

α2+aα+b=0.....(1)

α2+bα+a=0.....(2)

Subtracting eqn(1) from (2), we have

(α2+aα+b)−(α2+bα+a)=0

α2+aα+b−α2−bα−a=0

(a−b)α−(a−b)=0

(a−b)(α−1)=0

⇒α=1

Substituting the value of α in eqn(1), we have

(1)2+a(1)+b=0

⇒a+b+1=0

Hence the correct answer is (B)a+b+1=0

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