If f(x) = x2 + 5x + p and g(x) = x2 + 3x + q have a common factor then

1.	If f(x) = x2 + 5x + p and g(x) = x2 + 3x + q have a common factor then (p – q)2 = ……………A) 2(3p – 5q) B) 5p – 3q C) 3p – 5q D) 2(5p – 3q)
Answer» Correct option is (A) 2(3p – 5q) Let \((x-\alpha)\) be common factor of f(x) and g(x). \(\therefore\) \(x=\alpha\) be common zero of f(x) and g(x). \(\therefore\) \(\alpha^2+5\alpha+p=0\) and \(\alpha^2+3\alpha+q=0\) \(\Rightarrow\) \((\alpha^2+5\alpha+p)\) \(-(\alpha^2+3\alpha+q)=0\) \(\Rightarrow\) \(2\alpha+p-q=0\) \(\Rightarrow\) \(\alpha=\frac{q-p}2\) is a common root of \(\alpha^2+5\alpha+p=0\) \(\therefore\) \((\frac{q-p}2)^2+5(\frac{q-p}2)+p=0\) \(\Rightarrow\) \((q-p)^2+10(q-p)+4p=0\) \(\Rightarrow\) \((q-p)^2+10q-6p=0\) \(\Rightarrow\) \((q-p)^2=6p-10q\) = 2 (3p – 5q) Correct option is A) 2(3p – 5q)

If f(x) = x2 + 5x + p and g(x) = x2 + 3x + q have a common factor then (p – q)2 = ……………A) 2(3p – 5q) B) 5p – 3q C) 3p – 5q D) 2(5p – 3q)

Answer»

Correct option is (A) 2(3p – 5q)

Let \((x-\alpha)\) be common factor of f(x) and g(x).

\(\therefore\) \(x=\alpha\) be common zero of f(x) and g(x).

\(\therefore\) \(\alpha^2+5\alpha+p=0\) and \(\alpha^2+3\alpha+q=0\)

\(\Rightarrow\) \((\alpha^2+5\alpha+p)\) \(-(\alpha^2+3\alpha+q)=0\)

\(\Rightarrow\) \(2\alpha+p-q=0\)

\(\Rightarrow\) \(\alpha=\frac{q-p}2\) is a common root of \(\alpha^2+5\alpha+p=0\)

\(\therefore\) \((\frac{q-p}2)^2+5(\frac{q-p}2)+p=0\)

\(\Rightarrow\) \((q-p)^2+10(q-p)+4p=0\)

\(\Rightarrow\) \((q-p)^2+10q-6p=0\)

\(\Rightarrow\) \((q-p)^2=6p-10q\) = 2 (3p – 5q)

Correct option is A) 2(3p – 5q)