Question

Let $f(x)$ and $g(x)$ are polynomials of degree 3, where $g(\alpha)=g'(\alpha)=0, g''(\alpha) \neq 0$ and $\lim_{x \to \alpha} \frac{f(x)}{g(x)}=0$. If $h(x)=f(x)g'(x)+f'(x)\cdot g(x)$, then which of the following is/are true?

• A. Number of solution(s) which are common to all $f(x)=0, g(x)=0, h(x)=0$ is 1.
• B. Number of solution(s) which are common to all $f(x)=0, g(x)=0, h(x)=0$ is 2.
• C. Number of distinct roots of equation $h(x)=0$ is 2.
• D. $h'''(\alpha)=0$

Answer 1

Answer: (D)

h'''(α) = 0

Answer 2

Since $g(x)$ has a double zero and $f(x)$ a triple zero at $\alpha$, $f(x)=c(x–\alpha)^3$ and $g(x)= d(x–\alpha)^2(x–\beta)$. Then $h(x) = (f(x)g(x))' = c \cdot d \cdot (x–\alpha)^4[6x – (5\beta+\alpha)]$, so the only common zero is $x = \alpha$, $h(x)=0$ has two distinct roots, and because of the $(x–\alpha)^4$ factor, $h'''(\alpha)=0$.