Factor & remainder theorem

It is given that

P(x) = 4x³ + 8x² + 11x + 4

Use the factor theorem to show that (2x + 1) is a factor of P(x)

Express P(x) in the form

P (x) = (2 x + 1)(ax 2 + bx + c)

where a, b and c are constants to be found.

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.

Given that

use the factor theorem to prove that (5x 1) is a factor of P(x).

Factorise P (x) completely.

Hence, prove that 250n³ 300n² 110n 12 is a multiple of 12 when n is a positive whole number.

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.

p(x) 30x³ 7x² 7x 2

Prove that (2x 1) is a factor of p(x)

Factorise p (x) completely.

Prove that there are no real solutions to the equation