Algebraic division

Use the factor theorem to show that x 5 is a factor of 2x³ 9x² 11x 30

Hence, factorise completely 2x³ 9x² 11x 30

Show clear algebraic working.

x 4 is a factor of 6x³ kx² 26x 24, where k is a constant.

Use the factor theorem to show that k 19

Hence solve 6x³ 19x² 26x 24 0

Show clear algebraic working.

28

Use the factor theorem to show that (3x 4) is not a factor of f(x)

Factorise fully g(x)

Show clear algebraic working.

f (x) 3x³ ax² 20x b where a and b are integers.

(x + 4) is a factor of f(x)

(x 2) is a factor of f(x)

Use the factor theorem to find the value of a and the value of b

One solution of the equation 8x³ 18x² 5x 6 0 is

Without using a calculator and showing all your working, find the other 2 solutions of the equation.

Give your answers in exact form.

21(x + 2) is a factor of 6x³ + 31x² + kx + 30

Use the factor theorem to show that k = 53

Factorise fully 6x³ + 31x² + 53x + 30

Use the factor theorem to show that x 1 is a factor of 18x³ 9x² 17x 10

Factorise fully 18x³ 9x² 17x 10

Show clear algebraic working.

f(x) = 2x³ + 9x² – 14x – 9

Using the factor theorem, show that (2x – 1) is not a factor of f(x)

where a and b are integers to be found.

Hence find the exact solutions of f(x) = 0

The determinant of matrix A is equal to the determinant of matrix B

Show that k is a root of the equation

Given that f(k) = 2k³ + 5k² – 28k – 15

use the factor theorem to show that (k + 5) is a factor of f(k)

Hence, factorise completely f(k)

The transformation with matrix C, where C is a 22 matrix, is equivalent to the transformation with matrix A followed by the transformation with matrix B

Given that k is positive,

find matrix C

use the factor theorem to show that k = 23

Solve the equation 3x³ − 20x² + 23x + 10 = 0

Show clear algebraic working.

f(x) = 3x³ – 7x² + 5x – 1

Use the factor theorem to show that (3x – 1) is a factor of f(x).

Hence solve the equation f(x) = 0

Show clear algebraic working.

The curve C has equation y = f(x)

Find the exact coordinates of the turning points on C.

The curve C crosses the y-axis at the point P.

Find the gradient of C at P.

Hence find an equation of the tangent to C at P.

f (x) = 4x³ – 13x – 6

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

Hence factorise f (x) fully.

The curve C has equation y = f (x)

Find the coordinates of the points of intersection of C with the x‑axis.

Find the coordinates, to 2 decimal places, of the turning points of C.

The table below gives the coordinates of three points on C.

On the grid opposite, draw the curve C for –2 x 2

Clearly label the coordinates of the turning points of C and the coordinates of the points of intersection with the x ‑axis and the y ‑axis.

Turn over for a spare grid if you need to redraw your curve.

Only use this grid if you need to redraw your curve.

(x + 2) is a factor of x³ – 3x² + ax + 12 where a is a constant.

Show that a = –4

Factorise completely x³ – 3x² – 4x + 12