Theory question 111
Polynomials(a) Express 9x/((2x + 1)(x2 + 1)) in partial fractions.
(b) If 2mP2 − 10 = mP2, find the positive value of m.
Show the model answer
Model answer
(a) 9x/((2x + 1)(x² + 1)) = A/(2x + 1) + (Bx + C)/(x² + 1); 9x = A(x² + 1) + (Bx + C)(2x + 1).
x = −½: −9/2 = (5/4)A, A = −18/5. x² terms: A + 2B = 0, B = 9/5. Constants: A + C = 0, C = 18/5.
Answer: −18/(5(2x + 1)) + (9x + 18)/(5(x² + 1)).
(b) 2m(2m − 1) − 10 = m(m − 1), so 3m² − m − 10 = 0, (3m + 5)(m − 2) = 0. m = 2.
Explanation
A quadratic factor that doesn't factorise needs a linear numerator Bx + C.