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.
303 exam-style questions on OCR (MEI) A Level Maths 1.2 Algebra, covering 1.2.1 Algebraic vocabulary and notation, 1.2.2 Solve linear equations, 1.2.3 Change the subject of a formula, 1.2.4 Solve quadratic equations, 1.2.5 Discriminant of a quadratic, 1.2.6 Linear simultaneous equations, 1.2.7 One linear one quadratic simultaneous equations, 1.2.8 Points of intersection and solutions, 1.2.9 Linear inequalities, 1.2.10 Quadratic inequalities, 1.2.11 Expressing solutions of inequalities, 1.2.12 Use and manipulate surds, 1.2.13 Rationalise the denominator, 1.2.14 Laws of indices, 1.2.15 Negative, fractional and zero indices, 1.2.16 Proportional relationships, 1.2.17 Partial fractions (A-level only), and 1.2.18 Simplify rational expressions (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.