Algebraic Methods
Calculators may NOT be used to answer these questions unless a
the question.
1. Simplify
symbol is shown next to
x2  7 x  8
( x  8) 2
(Total for Question 1 is 2 marks)
2. Express
2x
7
as a single fraction.

x3 x3
Give your answer in its simplest form.
(Total for Question 2 is 3 marks)
3. Write
x
x 3

as a single fraction.
3x  4 2 x  1
Give your answer in its simplest form.
(Total for Question 3 is 3 marks)
x2  9
4. Simplify fully
( x  3)2 ( x  3) 2
(Total for Question 4 is 2 marks)
5. Simplify
x2  9
x2  4x  3
(Total for Question 5 is 3 marks)
6. Simplify
u2
u  4u  4
2
(Total for Question 6 is 2 marks)
7. Write
x
x2

as a single fraction.
x3 x3
Give your answer in its simplest form.
(Total for Question 7 is 3 marks)
8. Simplify fully
x 1 x2  x

x  7 x 2  49
(Total for Question 8 is 3 marks)
9. Express
x3
4 x

as a single fraction.
2x  3 2x  3
Give your answer in its simplest form.
(Total for Question 9 is 3 marks)
10. Simplify
6 x  10
6 x  22 x  20
2
(Total for Question 10 is 2 marks)
11. Write
8
x
as a single fraction.
 2
5x  5 y x  y 2
Give your answer in its simplest form.
(Total for Question 11 is 4 marks)
12. Simplify
3
9x
 2
2x 6x  4x
(Total for Question 12 is 3 marks)
13. Simplify
x2  2 x
x 2  7 x  10
(Total for Question 15 is 2 marks)
14.
Simplify fully
3x  x 2 2 x  6

x 2  9 ( x  3)2
(Total for Question 14 is 4 marks)
x3  3x 2  4 x  12
B
can be written in the form A + , where A and B
3
2
x
x  5x  6 x
are integers to be found.
15. Show that
(Total for Question 13 is 3 marks)
16. f(x) = 2x³ – 7x² – 17x + 10. Use the factor theorem and division to factorise f(x)
completely.
(Total for Question 16 is 6 marks)
g(x) = 4x³ – 8x² – 35x + 75
17.
(a) Use the factor theorem to show that (x + 3) is a factor of g(x)
(2)
(b) Hence show that g(x) can be written in the form g(x) = (x + 3) (ax + b) ²,
where a and b are constants to be found.
(4)
(Total for Question 17 is 6 marks)
18.
f(x) = x³ + 6x² + px + q
Given that f(4) = 0 and f(-5) = 36
(3)
(a) Find the values of p and q
(b) Factorise f(x) completely.
(4)
(Total for Question 18 is 7 marks)
19.
f(x) = x³ + kx – 2
(a) Given that (x – 2) is a factor of f(x) find the value of k
(2)
(b) Solve the equation f(x) = 0
(4)
(Total for Question 19 is 6 marks)
20.
f(x) = x 3 + 6x 2 + 4x – 15
(a) Use the factor theorem to show that x = –3 is a solution to f(x) = 0
(2)
(b) Find the other solutions to the equation f(x) = 0 giving your answers to 2 dp.
(4)
(Total for Question 20 is 7 marks)