An engineering student, Mia, is analyzing a system transfer function H(s)=5s2−2(s+2)(s−1)2H(s) = \frac{5s^2 - 2}{(s + 2)(s - 1)^2}H(s)=(s+2)(s−1)25s2−2. She attempts to decompose it into partial fractions for an inverse Laplace transform calculation.
Her incorrect attempt is shown below.
Step 1: 5s2−2(s+2)(s−1)2≡Ps+2+Q(s−1)2\frac{5s^2 - 2}{(s + 2)(s - 1)^2} \equiv \frac{P}{s + 2} + \frac{Q}{(s - 1)^2}(s+2)(s−1)25s2−2≡s+2P+(s−1)2Q
Step 2: 5s2−2≡P(s−1)2+Q(s+2)5s^2 - 2 \equiv P(s - 1)^2 + Q(s + 2)5s2−2≡P(s−1)2+Q(s+2)
Step 3: Let s=−2⇒18=9P⇒P=2s = -2 \Rightarrow 18 = 9P \Rightarrow P = 2s=−2⇒18=9P⇒P=2 Let s=1⇒3=3Q⇒Q=1s = 1 \Rightarrow 3 = 3Q \Rightarrow Q = 1s=1⇒3=3Q⇒Q=1
Mia's Result: 5s2−2(s+2)(s−1)2≡2s+2+1(s−1)2\frac{5s^2 - 2}{(s + 2)(s - 1)^2} \equiv \frac{2}{s + 2} + \frac{1}{(s - 1)^2}(s+2)(s−1)25s2−2≡s+22+(s−1)21
(i) By using a counter-example, show that the result obtained by Mia cannot be correct.
(ii) Explain the conceptual mistake Mia made in Step 1.
Determine the correct partial fraction decomposition for H(s)H(s)H(s), with constant numerators.
Practise Edexcel A Level Maths Algebraic Methods with exam-style questions for A Level Maths. 261 questions covering 7.1 Algebraic Fractions, 7.2 Dividing Polynomials, 7.3 The Factor Theorem, 7.4 Mathematical Proof, and 7.5 Methods of Proof, 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.
