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Algebraic Methods

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Question 232

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+125 V(x) = x^3 + (k + 5)x^2 + 5(k + 5)x + 125 V(x)=x3+(k+5)x2+5(k+5)x+125

where k k\,k is a constant.

a.

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.

[2]
b.

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.

[5]
Markscheme

Algebraic Methods Questions

  1. A Level
  2. /Maths
  3. /Algebraic Methods

368 exam-style questions on Edexcel A Level Maths Algebraic Methods, covering 1.1 Proof by Contradiction, 1.2 Algebraic Fractions, 1.3 Partial Fractions, 1.4 Repeated Factors, and 1.5 Algebraic Division. Each one has a worked solution and a mark scheme showing where the marks go.

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