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.
532 exam-style questions on AQA A Level Maths 1.5 B: Algebra and functions, covering 1.5.1 Laws of indices, 1.5.2 Surds, 1.5.3 Quadratic functions, 1.5.4 Simultaneous equations, 1.5.5 Linear and quadratic inequalities, 1.5.6 Polynomials and rational expressions, 1.5.7 Graphs of functions and proportion, 1.5.8 Composite and inverse functions (A-level only), 1.5.9 Transformations of graphs, 1.5.10 Partial fractions (A-level only), and 1.5.11 Functions in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.