Sketch the curve with equation
y=x2(2x−a) y = x^2(2x - a) y=x2(2x−a)where a>0a > 0a>0.
The polynomial p(x)p(x)p(x) is given by
p(x)=x2(2x−a)−64 p(x) = x^2(2x - a) - 64 p(x)=x2(2x−a)−64It is given that x−8x - 8x−8 is a factor of p(x)p(x)p(x). Use the factor theorem to show that a=15a = 15a=15.
State the transformation which maps the curve with equation y=x2(2x−15)y = x^2(2x - 15)y=x2(2x−15) onto the curve with equation y=x2(2x−15)−64y = x^2(2x - 15) - 64y=x2(2x−15)−64.
The polynomial x2(2x−15)−64x^2(2x - 15) - 64x2(2x−15)−64 can be written as (x−8)(2x2+bx+c)(x - 8)(2x^2 + bx + c)(x−8)(2x2+bx+c). Without finding the values of bbb and ccc, use your knowledge of the transformation in part (b)(ii) and the sketch in part (a) to explain why
b2<8c b^2 < 8c b2<8c
