Additional Mathematics 4037: Factors of polynomials Topical Past Paper Questions

It is given that $ p(x) = ax^{3} - 7x^{2} - bx + 9 $, where a and b are constants. x - 3 is a factor of p(x). When $ p(x) $ is divided by $ x+2 $ the remainder is -35. Find the values of a and b.

DO NOT USE A CALCULATOR IN THIS QUESTION. The polynomial p is defined by $ p(x) = ax^{3} - 3x^{2} - 3x + b $, where a and b are constants. (a) Given that x = 2 and x = -1 are roots of the equation $ p(x) = 0 $, find a and b. (b) Solve the equation $ p(x)=0 $.

(a) Find the equation of the normal to the curve $ y = x^{3} - 7x^{2} + 12x - 5 $ at the point (1, 1). [5] (b) Find the x-coordinates of the two points where the normal cuts the curve again. Give your answers in the form $ x = a \pm \sqrt{b} $ where a and b are integers. [5]