Theory question 103
PolynomialsIf (x + 2) and (x − 1) are factors of f(x) = 6x4 + mx3 − 13x2 + nx + 14, find the; (a) values of m and n; (b) remainder when f(x) is divided by (x + 1).
Show the model answer
Model answer
(a) f(−2) = 0: 96 − 8m − 52 − 2n + 14 = 0, so 4m + n = 29
f(1) = 0: 6 + m − 13 + n + 14 = 0, so m + n = −7
Subtract: 3m = 36, m = 12, n = −19
(b) f(x) = 6x⁴ + 12x³ − 13x² − 19x + 14
Remainder = f(−1) = 6 − 12 − 13 + 19 + 14 = 14
Explanation
Use the factor theorem: f(a) = 0 when (x − a) is a factor; the remainder on dividing by (x + 1) is f(−1).