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Algebra Worksheet 2 - Factorising
1. Factorise the following by taking out common factors:
a) x 2 + 3x
b) x 2 - x -3
c) 4a -1 + 6a 4
3x 4
x3
d)
+
5 y 10 y
1
2
e) x - x
f) a
1
2
3
2
+a
-1
g)
h)
i)
j)
k)
l)
ab + ac + bd + cd
12bc + 35a 2 - 28ac - 15ab
3a 2 b(2s - t ) - 6ab(t - 2s) - 9ab 2 (2s - t )
4 x 3 ( x - y) 2 - 6x 2 y ( x - y ) 3
(a + b) 2 - 3a - 3b
2 x( x - 2) + xy - 2 y(3y + 4)
h)
i)
j)
k)
l)
( x + 5) 2 + 4( x + 5) + 3
(2a - 3) 2 - 8(2a - 3) + 16
2 g 2 h 2 - 8gh - 120
12 sin 2 x - 5 sin x cos x - 2 cos2 x
4 x - 23+x - 20
2
2. Factorise the following quadratic expressions:
a)
b)
c)
d)
e)
f)
g)
x 2 + 11x + 30
6x 2 - 84 x - 192
12 x 2 - x - 6
7 tan 2 x - 21 tan x - 196
cos3 x - 7 cos2 x + 12 cos x
x 4 - 2 x 2 - 24
x 6 - 2 x 3 - 35
3. Factorise the following using common prime factor, where possible:
a)
b)
c)
d)
e)
f)
8x 2 + 14 x - 15
9 x 2 + 27 x + 20
50x 2 + 75x - 27
36x 2 + 105x + 49
200 x 2 + 30x - 27
121x 2 - 55x - 50
g)
h)
i)
j)
k)
27 x 2 + 66x - 80
600 x 2 - 90x - 27
28x 2 - 77 x - 72
250 x 2 + 35x - 98
216 x 2 + 30x + 5
(Sealy, Agnew; Senior Mathematics p9) – see rules at the end of this worksheet
4. Factorise:
a) x 2 - y 2
b) 0.49 x 4 - 0.64 y 8
c) ( x + 3) 2 - ( x - 2) 2
d) (2 x + 3) 2 - ( x - 4) 2
e) 32 x - 4
x2
f) 1 25
3
2
g)
4 - 12 y
x
1
h) x 2 - 2 + 2
x
Delta: p 1-4, Ex 1.1-1.8
Mathematics with Calculus
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JM
Algebra Worksheet 2 - Factorising
(Sidebotham; Mathematics Revision p5)
Exercise 1.2
1. Factorise each of the following
a.
e.
i.
m.
x 2 - x -12
x 2 - x - 20
6x 2 + 7x - 5
8z 2 + 2zy -15y 2
p. x 2 +11ax + 28a2
x 2 -16
3x 2 + 4x +1
25y 2 - 36
t4 - t2
16a 2
q.
-1
b2
b.
f.
j.
n.
c.
g.
k.
o.
x 2 + 3x -18
4x 2 + 8x + 3
x 4 -16
t 4 -13t 2 + 36
d. a2bc - bc 2d
h. 6x 2 +17x - 3
l. 4x 2 y - 2y 3
2. Factorise as far as possible
a. ( x -1) - 2( x -1) +1
c. 2x 3 + x 2 - 8x - 4
e. 2a - 2b + a2 - b2
b. 1- (a + b)2
d. 3x - 3y + x 2 - xy
f. 4x 2 (a - b) - (a - b)
g. x 3 + 8
h.
i. 64y 3 - 27
k. 125y 3 + 64 x 3
*m. 24 x -1
*j. ( x - y ) - 4(x - y)(x + y)2
l. sin4 q cos2 q + sinq cosq - 2
2
3
1
( x +1) 2 - ( x +1) 2
3
3. Simplify each of the following by factorising
a.
c.
x2 - 4
x 2 + 3x -10
x3 - 8
x2 - 4
1
a 2b + 2ab 2
ab
2 2x + 2 x +1 - 3
d.
2x + 3
b.
1
4(x - 3) 2 - (2x + 3)(x - 3) 2
e.
x-3
Rules:
Rule 1. (Most important) Always make sure that you have taken out all common factors first.
6
Rule 2. No horizontal row may contain a common factor. For example 32 -1
is not a possible
arrangement, since the top row has a common factor of 3.
Rule 3. If the middle term and one of the end terms have a common prime factor, then that prime
factor must be a common factor of the two numbers in the corresponding vertical column. As 3 is a
common factor of 18x 2 and - 3x in our example, then, 18 must be split into 63 with the common
prime factor 3.
Note: This rule applies to prime numbers. If 4 (a perfect square) is a common factor of the middle
and one of the end terms, you can only be sure that the end term splits into two numbers with 2 as a
common factor. However, the rule does work for a number like 6. Why?
Rule 4. If a particular prime number p is a common factor of the middle term and an end term then
p2 must also be a factor of that end term. Otherwise the trinomial has no real factors. 6x 2 + 3x - 28
has no real factors, because 3 is a common factor of the first two terms, but 9 is not a factor of 6.
Mathematics with Calculus
Page 2
JM