Question

The above picture are few natural examples of parabolic shape which is represented by a quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.

Based on the above information, answer the following questions.

If $\alpha$ and $\beta$ are the zeros of polynomial
$x^2+3x-2$, find $\frac{1}{(\alpha {)}^3}+\frac{1}{(\beta {)}^3}$.

• A. $\frac{8}{45}$
• B. $\frac{-8}{45}$
• C. $\frac{45}{8}$
• D. $\frac{45}{8}$

Answer 1

The correct option is C $\frac{45}{8}$

Given polynomial is :
$x^2+3x-2$

$\Rightarrow \alpha +\beta =\frac{-b}{a}=\frac{-3}{1}=-3$

$\Rightarrow \alpha \beta =\frac{c}{a}=\frac{-2}{1}=-2$

$\therefore \frac{1}{(\alpha {)}^3}+\frac{1}{(\beta {)}^3}$

$=\frac{(\alpha {)}^3+(\beta {)}^3}{(\alpha \beta {)}^3}$

$=\frac{(\alpha +\beta {)}^3-3\alpha \beta (\alpha +\beta )}{(\alpha \beta {)}^3}$

$=\frac{(-3{)}^3-3(-2\left)\right(-3)}{(-2{)}^3}$

[Substituting the values of $\alpha +\beta ,\alpha \beta$]

$=\frac{-27-18}{-8}$

$=\frac{45}{8}$