1.2.8 Algebra and Functions

The potential energy $V (x)$ of a mechanical system as a function of displacement $x\,$ is modeled by the cubic polynomial

V(x) = x^3 + (k + 5)x^2 + 5(k + 5)x + 125

where $k\,$ is a constant.

Use the factor theorem to prove that $(x + 5)$ is a factor of $V (x)$ for all values of $k$ .

[2]

The graph of $y = V (x)$ meets the $x$ -axis at exactly two distinct points.

(i) Sketch a possible graph of $y = V (x)$ , showing the coordinates of the intercepts with both axes.

(ii) Given that $V (x)$ can be written in the form $V(x) = (x + 5)(x^2 + kx + 25)$ , determine the value of $k$ . Fully justify your answer.

[5]

1.2.8 Algebra and Functions Questions