A student attempts to write x2+5x(x−1)(x+4)2\displaystyle \frac{x^2 + 5x}{(x-1)(x+4)^2}(x−1)(x+4)2x2+5x in partial fractions with constant numerators.
Their incorrect attempt is shown below:
Step 1 x2+5x(x−1)(x+4)2≡Ax−1+B(x+4)2\displaystyle \frac{x^2 + 5x}{(x-1)(x+4)^2} \equiv \frac{A}{x-1} + \frac{B}{(x+4)^2}(x−1)(x+4)2x2+5x≡x−1A+(x+4)2B
Step 2 x2+5x≡A(x+4)2+B(x−1)x^2 + 5x \equiv A(x+4)^2 + B(x-1)x2+5x≡A(x+4)2+B(x−1)
Step 3 Let x=1⇒A=0.24x = 1 \Rightarrow A = 0.24x=1⇒A=0.24 Let x=−4⇒B=0.8x = -4 \Rightarrow B = 0.8x=−4⇒B=0.8
Answer 0.24x−1+0.8(x+4)2\displaystyle \frac{0.24}{x-1} + \frac{0.8}{(x+4)^2}x−10.24+(x+4)20.8
Explain the mistake that the student has made in step 1.
