Theory question 111
Polynomials(a) Express (2x2 − 5x + 1)/(x3 − 4x2 + 3x) in partial fractions.
(b) If |x − 3 −4 3|
|5 2 2|
|2 −4 6 − x| = −24, find the value of x.
Show the model answer
Model answer
(a) x3 − 4x2 + 3x = x(x − 1)(x − 3)
A/x + B/(x − 1) + C/(x − 3): A = 1/3, B = 1, C = 2/3
= 1/(3x) + 1/(x − 1) + 2/(3(x − 3))
(b) Expanding along the first row: (x − 3)(20 − 2x) + 4(26 − 5x) + 3(−24) = −2x2 + 6x − 28
−2x2 + 6x − 28 = −24, x2 − 3x + 2 = 0
x = 1 or x = 2
Explanation
Factorise the denominator fully before splitting; expand the determinant along the first row.