Algebraic division

Given that

• f(x) = 2x³ + 3x² – 16x + 16

• f(–4) = 0

write f(x) in the form (x + a) Q (x) where a is a constant and Q(x) is a quadratic expression.

Hence prove that –4 is the only real root of the equation f(x) = 0 (Solutions relying on calculator technology are not acceptable.)

Figure 3 shows a sketch of the curve C with equation

and the straight line l with equation y = 2 x – 10

Verify that C and l intersect where x = 6 The curve and line also intersect at the point Q shown in Figure 3.

Show that the x coordinate of Q is a solution of

Using algebra and showing all stages of working, find the exact x coordinate of Q.

The curve C has equation

Find an expression for in terms of x and y.

The point P(1, 0) lies on C.

Show that the normal to C at P has equation

Prove that the normal to C at P does not meet C again.

You should use algebra for your proof and make your reasoning clear.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

where a is a positive constant.

Given (x – a) is a factor of f(x),

show that

Hence

find the value of a

use algebra to find the exact solutions of the equation

f(x) = 3

The function f is defined by

Show that

where A and B are constants to be found. The function g is defined by

State the range of g⁻¹

Find the range of f g⁻¹

Calculate f (2)

Write f (x) as a product of two algebraic factors.

Using the answer to (a)(ii),

prove that there are exactly two real solutions to the equation

deduce the number of real solutions, for 7π θ <10π, to the equation

Given that

find the values of the constants a, b, c, d and e.

f(x) = 24x³ +Ax² − 3x+B where A and B are constants. When f(x) is divided by (2x− 1) the remainder is 30

Show that A+ 4B= 114 Given also that (x+ 1) is a factor of f(x),

find another equation in A and B.

Find the value of A and the value of B.

Hence find a quadratic factor of f(x).

f(x) = –6x³ – 7x² + 40x + 21

Use the factor theorem to show that (x + 3) is a factor of f(x)

Factorise f(x) completely.

Hence solve the equation

giving your answer to 2 decimal places.

f(x) = 6x³ + 13x² – 4

Use the remainder theorem to find the remainder when f(x) is divided by (2 x + 3).

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

Factorise f(x) completely.

f(x) = 2x³ – 7x² + 4x + 4

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

Factorise f(x) completely.

f(x) = 2x³ – 5x² + ax + 18 where a is a constant. Given that (x – 3) is a factor of f(x),

show that a = – 9

factorise f(x) completely.

Given that

find the values of y that satisfy g(y) = 0, giving your answers to 2 decimal places where appropriate.

f(x) = ax³ – 11x² + bx + 4, where a and b are constants. When f(x) is divided by (x – 3) the remainder is 55 When f(x) is divided by (x + 1) the remainder is –9

Find the value of a and the value of b.

Given that (3 x + 2) is a factor of f(x),

factorise f(x) completely.

f(x) = 2x³ – 7x² – 10x + 24

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

Factorise f(x) completely.

f (x) = 2x³ − 7x² − 5x + 4

Find the remainder when f (x) is divided by (x –1).

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

Factorise f (x) completely.

₃₂

Use the factor theorem to show that (x + 4) is a factor of f (x).

Factorise f (x) completely.