The discriminant

Sketch the curve with equation

The polynomial p (x) is given by

It is given that x 3 is a factor of p(x)

Use the factor theorem to show a 2

State the transformation which maps the curve with equation

onto the curve with equation

The polynomial x²(2x 2) 36 can be written as (x 3)(2x² bx c)

Without finding the values of b and c, use your answers to parts (a) and (b)(ii) to explain why

The polynomial p (x) is given by

where b is a constant.

Use the factor theorem to prove that (x 2) is a factor of p(x) for all values of b.

The graph of y p(x) meets the x-axis at exactly two points.

Sketch a possible graph of y p(x)

Given p (x) can be written as

find the value of b.

Fully justify your answer.

Four possible sketches of y ax² bx c are shown below.

Given b² 4ac 0 and a, b and c are non-zero constants, which sketch is the only one that could possibly be correct?

Tick () one box.

p(x) 4x³ 15x² 48x 36

Use the factor theorem to prove that x 6 is a factor of p(x).

Prove that the graph of y p(x) intersects the x-axis at exactly one point.

State the coordinates of this point of intersection.

A curve, C, has equation y x² 6x k, where k is a constant.

The equation x² 6x k 0 has two distinct positive roots.

Sketch C on the axes below.

Find the range of possible values for k.

Fully justify your answer.