Theory question 110
Polynomials(a) If f(x) = (2x − 3)/((x2 − 1)(x − 2)), (i) find the values of x for which f(x) is undefined. (ii) express f(x) in partial fractions.
(b) A circle with centre (−3, 1) passes through the point (3, 1). Find its equation.
Show the model answer
Model answer
(a)(i) x = 1, −1 or 2.
(ii) A/(x − 1) + B/(x + 1) + C/(x − 2): x = 1 gives A = ½; x = −1 gives B = −5/6; x = 2 gives C = ⅓.
f(x) = 1/(2(x − 1)) − 5/(6(x + 1)) + 1/(3(x − 2)).
(b) r = 6: (x + 3)² + (y − 1)² = 36, i.e. x² + y² + 6x − 2y − 26 = 0.
Explanation
Factorise x² − 1 first.