A student attempts to write 2x2−x+1(x+1)(x−1)2\displaystyle \frac{2x^2 - x + 1}{(x+1)(x-1)^2}(x+1)(x−1)22x2−x+1 in partial fractions with constant numerators.
Their incorrect attempt is shown below:
Step 1 2x2−x+1(x+1)(x−1)2≡A(x+1)+B(x−1)2\displaystyle \frac{2x^2 - x + 1}{(x+1)(x-1)^2} \equiv \frac{A}{(x+1)} + \frac{B}{(x-1)^2}(x+1)(x−1)22x2−x+1≡(x+1)A+(x−1)2B
Step 2 2x2−x+1≡A(x−1)2+B(x+1)2x^2 - x + 1 \equiv A(x-1)^2 + B(x+1)2x2−x+1≡A(x−1)2+B(x+1)
Step 3 Let x=1⇒B=1x = 1 \Rightarrow B = 1x=1⇒B=1 Let x=−1⇒A=1x = -1 \Rightarrow A = 1x=−1⇒A=1
Answer 1(x+1)+1(x−1)2\displaystyle \frac{1}{(x+1)} + \frac{1}{(x-1)^2}(x+1)1+(x−1)21
Explain the mistake that the student has made in step 1.
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.