A student attempts to write 3x2−30x(x+2)(x−4)2\displaystyle \frac{3x^2 - 30x}{(x+2)(x-4)^2}(x+2)(x−4)23x2−30x in partial fractions with constant numerators.
Their incorrect attempt is shown below:
Step 1 3x2−30x(x+2)(x−4)2≡A(x+2)+B(x−4)2\displaystyle \frac{3x^2 - 30x}{(x+2)(x-4)^2} \equiv \frac{A}{(x+2)} + \frac{B}{(x-4)^2}(x+2)(x−4)23x2−30x≡(x+2)A+(x−4)2B
Step 2 3x2−30x≡A(x−4)2+B(x+2)3x^2 - 30x \equiv A(x-4)^2 + B(x+2)3x2−30x≡A(x−4)2+B(x+2)
Step 3 Let x=4⇒B=−12x = 4 \Rightarrow B = -12x=4⇒B=−12 Let x=−2⇒A=2x = -2 \Rightarrow A = 2x=−2⇒A=2
Answer 2(x+2)−12(x−4)2\displaystyle \frac{2}{(x+2)} - \frac{12}{(x-4)^2}(x+2)2−(x−4)212
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.