Polynomials

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 1 shows a sketch of a curve C with equationy = f(x), where f(x) is a quartic expression in x. The curve

• has maximum turning points at (–1, 0) and (5, 0)

• crosses the y ‑axis at (0, –75)

• has a minimum turning point at x = 2

Find the set of values of x for which

writing your answer in set notation.

Find the equation of C. You may leave your answer in factorised form. The curve C₁ has equation y = f(x) + k, where k is a constant. Given that the graph of C₁ intersects the x‑axis at exactly four places,

find the range of possible values for k.

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.

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Factorise completely

g(x) = 3x³ 20x² k 17 x k

where k is a constant.

Given that (x – 3) is a factor of g(x), find the value of k.

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

f (x) = ax³ + 10x² − 3ax − 4 Given that (x − 1) is a factor of f (x), find the value of the constant a. You must make your method clear.

f (x) = ax³ + 10x² − 3ax − 4

Given that (x − 1) is a factor of f (x), find the value of the constant a.

You must make your method clear.

f(x) = 3x³ + 2ax² – 4x + 5a

Given that (x + 3) is a factor of f(x), find the value of the constant a.

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

Factorise completely 9x – 4x³

Sketch the curve C with equation

Show on your sketch the coordinates at which the curve meets the x-axis.

The points A and B lie on C and have x coordinates of –2 and 1 respectively.

Show that the length of AB is k 10 where k is a constant to be found.

Factorise fully 25x – 9x³

The curve C passes through the point (—1, 0) and touches the x-axis at the point (2, 0). The curve C has a maximum at the point (0, 4).

The equation of the curve C can be written in the form

where a, b andc are integers. Calculate the values of a, b and c. (5)

Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes. (3)

Factorise completely x − 4x³

Factorise completely x³ 4x

Sketch the curve C with equation

showing the coordinates of the points at which the curve meets the x-axis.

The point A with x-coordinate 1 and the point B with x-coordinate 3 lie on the curve C.

Find an equation of the line which passes through A and B, giving your answer in the form y = mx + c, where m and c are constants.

Show that the length of AB is k 10, where k is a constant to be found.

Factorise completely x³ – 6x² + 9x

Sketch the curve with equation

showing the coordinates of the points at which the curve meets the x-axis.

Using your answer to part (b), or otherwise,

sketch, on a separate diagram, the curve with equation

showing the coordinates of the points at which the curve meets the x-axis.

The point P (1, a) lies on the curve with equation y = (x + 1)²(2 – x).

Find the value of a.

On the axes below sketch the curves with the following equations:

On your diagram show clearly the coordinates of any points at which the curves meet the axes.

With reference to your diagram in part (b) state the number of real solutions to the equation

Factorise completely

The curve C has equation

Sketch C showing clearly the coordinates of the points where the curve meets the coordinate axes.

Show that the equation of C can be written in the form

where k is a positive integer, and state the value of k.

There are two points on C where the gradient of the tangent to C is equal to 3.

Find the x-coordinates of these two points.