Theory question 111
Binomial Theorem(a)(i) Write down the binominal expansion of (2 − ½x)5 in ascending powers of x. (ii) Using the expansion in 11(a)(i), find, correct to two decimal places, the value of (1.99)5.
(b) The polynomial x3 + qx2 + rx + 9, where q and r are constants, has (x + 1) as a factor and has a remainder −17 when divided by (x + 2). Find the values of q and r.
Show the model answer
Model answer
(a)(i) 32 − 40x + 20x² − 5x³ + (5/8)x⁴ − (1/32)x⁵.
(ii) 2 − ½x = 1.99 gives x = 0.02: 32 − 0.8 + 0.008 − 0.00004 + … ≈ 31.21.
(b) f(−1) = 0: −1 + q − r + 9 = 0, so q − r = −8.
f(−2) = −17: −8 + 4q − 2r + 9 = −17, so 2q − r = −9.
q = −1, r = 7.
Explanation
Choose x so that the bracket equals 1.99.