The potential energy V(x)V(x)V(x) of a mechanical system as a function of displacement x x\,x is modeled by the cubic polynomial
V(x)=x3+(k+5)x2+5(k+5)x+125V(x) = x^3 + (k + 5)x^2 + 5(k + 5)x + 125V(x)=x3+(k+5)x2+5(k+5)x+125
where k k\,k is a constant.
Use the factor theorem to prove that (x+5)(x + 5)(x+5) is a factor of V(x)V(x)V(x) for all values of kkk.
The graph of y=V(x)y = V(x)y=V(x) meets the xxx-axis at exactly two distinct points.
(i) Sketch a possible graph of y=V(x)y = V(x)y=V(x), showing the coordinates of the intercepts with both axes.
(ii) Given that V(x)V(x)V(x) can be written in the form V(x)=(x+5)(x2+kx+25)V(x) = (x + 5)(x^2 + kx + 25)V(x)=(x+5)(x2+kx+25), determine the value of kkk. Fully justify your answer.
Practise Edexcel A Level Maths 1.5 Algebraic Division with exam-style questions for A Level Maths. 65 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.