Theory question 109
Polynomials(a) The polynomial f(x) = x3 + px2 − 10x + q is exactly divisible by (x2 + x − 6). Find the: (i) values of p and q; (ii) third factor.
(b) The volume of a cube is increasing at the rate of 2½cm3s−1. Find the rate of change of the side of the base when its length is 2cm.
Show the model answer
Model answer
(a)(i) x2 + x − 6 = (x + 3)(x − 2), so f(2) = 0 and f(−3) = 0
f(2) = 8 + 4p − 20 + q = 0: 4p + q = 12
f(−3) = −27 + 9p + 30 + q = 0: 9p + q = −3
5p = −15: p = −3, q = 24
(ii) x3 − 3x2 − 10x + 24 = (x2 + x − 6)(x − 4); third factor (x − 4)
(b) V = s3, dV/dt = 3s2(ds/dt)
2.5 = 3(2)2(ds/dt), ds/dt = 2.5/12 = 5/24 ≈ 0.21 cm s−1
Explanation
Each factor of the divisor gives a zero of f(x).