Theory question 109
Differentiation(a) Differentiate (x − 3)(x2 + 5) with respect to x.
(b) If (x + 1)2 is a factor of f(x) = x3 + ax2 + bx + 3, where a and b are constants, find the: (i) values of a and b; (ii) zeros of f(x).
Show the model answer
Model answer
(a) d/dx = (x² + 5) + 2x(x − 3) = 3x² − 6x + 5.
(b)(i) f(x) = (x + 1)²(x + 3) (the constant term 3 fixes the third factor) = x³ + 5x² + 7x + 3, so a = 5, b = 7.
(ii) Zeros: −1 (twice) and −3.
Explanation
Compare coefficients after writing f(x) = (x + 1)²(x + c).