Question

If $a , b , c , d$ are real numbers such that $\frac{ a +2 c }{ b +3 d }+\frac{4}{3}=0$, then the equation $ax ^3+ bx x ^2+ cx + d =0$ has

• A. at least one root in $(-1,0)$
• B. at least one root in $(0,1)$
• C. at least two roots in $(-1,1)$
• D. no root in $(-1,1)$

Answer 1

Answer: (B)

$\frac{a+2 c}{b+3 d}+\frac{4}{3}=0 \Rightarrow 3 a+4 b+6 c+12 d=0$
$\int\limits_0^1\left(a x^3+b x^2+c x+d\right) d x=\frac{1}{12}(3 a+4 b+6 c+12 d)=0 .$
Hence, $ax ^3+ bx ^2+ cx + d =0$ has at least one root in $(0,1)$.