Factor & remainder theorem

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.

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

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

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) = ax³ + bx² 4x 3, where a and b are constants. Given that (x – 1) is a factor of f(x),

show that a + b = 7

Given also that, when f(x) is divided by (x + 2), the remainder is 9,

find the value of a and the value of b, showing each step in your working.

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.

f (x) 3x − 2 x − k − 8 where k is a constant.

Write down the value of f (k).

When f (x) is divided by x − 2 the remainder is 4

Find the value of k.

Factorise f (x) completely.

₃₂

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

Factorise f (x) completely.