Elena is attempting to decompose the rational function R(x)=5x2−10x+9(x+2)(x−1)2R(x) = \frac{5x^2 - 10x + 9}{(x + 2)(x - 1)^2}R(x)=(x+2)(x−1)25x2−10x+9 as part of a stress-analysis calculation for a suspension bridge.
Her initial calculation is shown below.
Step 1: 5x2−10x+9(x+2)(x−1)2≡Ax+2+B(x−1)2\frac{5x^2 - 10x + 9}{(x + 2)(x - 1)^2} \equiv \frac{A}{x + 2} + \frac{B}{(x - 1)^2}(x+2)(x−1)25x2−10x+9≡x+2A+(x−1)2B
Step 2: 5x2−10x+9≡A(x−1)2+B(x+2)5x^2 - 10x + 9 \equiv A(x - 1)^2 + B(x + 2)5x2−10x+9≡A(x−1)2+B(x+2)
Step 3: Let x=1⇒4=3B⇒B=43x = 1 \Rightarrow 4 = 3B \Rightarrow B = \frac{4}{3}x=1⇒4=3B⇒B=34 Let x=−2⇒49=9A⇒A=499x = -2 \Rightarrow 49 = 9A \Rightarrow A = \frac{49}{9}x=−2⇒49=9A⇒A=949
Result: 5x2−10x+9(x+2)(x−1)2≡499(x+2)+43(x−1)2\frac{5x^2 - 10x + 9}{(x + 2)(x - 1)^2} \equiv \frac{49}{9(x + 2)} + \frac{4}{3(x - 1)^2}(x+2)(x−1)25x2−10x+9≡9(x+2)49+3(x−1)24
(i) By using a counter-example, show that Elena's result is incorrect.
(ii) Explain the error Elena made in Step 1.
Express 5x2−10x+9(x+2)(x−1)2\frac{5x^2 - 10x + 9}{(x + 2)(x - 1)^2}(x+2)(x−1)25x2−10x+9 as partial fractions with constant numerators.
