​ If (x – 4) is the HCF of p(x) = x^2 – nx -12 and q(x) = x^2 – mx – 8 then the simplest form of p(x)/q(x) is ……………… ​

Question

If (x – 4) is the HCF of p(x) = x² – nx -12 and q(x) = x² – mx – 8 then the simplest form of \(\frac{p(x)}{q(x)}\) is ………………

A) \(\frac{x+2}{x+3}\)

B) \(\frac{x+3}{x-2}\)

C) \(\frac{x+3}{x+2}\)

D) \(\frac{x-2}{x-3}\)

Answer 1

Correct option is (C) \(\frac{x+3}{x+2}\)

\(\because\) (x - 4) is the HCF of p(x) and q(x).

\(\therefore\) (x - 4) is a common factor of p(x) and q(x).

\(\therefore\) x = 4 is a common zero of p(x) and q(x).

\(\therefore\) 16 - 4n - 12 = 0 \(\Rightarrow\) 4n = 16 - 12 = 4 \(\Rightarrow\) n = 1

and 16 - 4m - 8 = 0 \(\Rightarrow\) 4m = 16 - 8 = 8 \(\Rightarrow\) m = 2

\(\therefore\) p(x) = \(x^2-x-12\) = (x - 4) (x + 3)

and q(x) = \(x^2-2x-8\) = (x - 4) (x + 2)

\(\therefore\) \(\frac{p(x)}{q(x)}=\frac{(x-4)(x+3)}{(x-4)(x+2)}=\frac{x+3}{x+2}\)

Answer 2

Correct option is B) \(\frac{x+3}{x-2}\)