Theory question 110
Polynomials(a) A quadratic polynomial, g(x) has (2x + 1) as a factor. If g(x) is divided by (x + 1) and (x − 2), the remainders are −6 and −5 respectively. Find: (i) g(x); (ii) the zeros of g(x).
(b) Find the third term when (x/2 − 1)8 is expanded in descending powers of x.
Show the model answer
Model answer
(a)(i) Let g(x) = (2x + 1)(ax + b). g(−1) = −(−a + b) = a − b = −6; g(2) = 5(2a + b) = −5, so 2a + b = −1.
a = −7/3, b = 11/3: g(x) = (2x + 1)(11 − 7x)/3 = (−14x² + 15x + 11)/3.
(ii) Zeros: x = −½ and x = 11/7.
(b) Third term = ⁸C₂(x/2)⁶(−1)² = 28x⁶/64 = 7x⁶/16.
Explanation
Use the remainder theorem: the remainder on dividing by (x − c) is g(c).