Theory question 111
Polynomials(a) Express (8x2 + 8x + 9)/((x − 1)(2x + 3)2) in partial fractions.
(b) The coordinates of the centre and circumference of a circle are (−2, 5) and 6π units respectively. Find the equation of the circle.
Show the model answer
Model answer
(a) Let it equal A/(x − 1) + B/(2x + 3) + C/(2x + 3)²; 8x² + 8x + 9 = A(2x + 3)² + B(x − 1)(2x + 3) + C(x − 1).
x = 1: 25 = 25A, A = 1. x = −3/2: 15 = −(5/2)C, C = −6. x² terms: 8 = 4A + 2B, B = 2.
Answer: 1/(x − 1) + 2/(2x + 3) − 6/(2x + 3)².
(b) 2πr = 6π gives r = 3. (x + 2)² + (y − 5)² = 9, i.e. x² + y² + 4x − 10y + 20 = 0.
Explanation
A repeated factor needs a fraction for each power of it.