Download 8 Expanding brackets and factorising

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Chapter 8 Expanding brackets and factorising
Unit 2
8 Expanding brackets and factorising
Key Points
expanding brackets: multiply each term inside
the bracket by the term outside the bracket.
factorising: the opposite of expanding brackets.
Find a common factor of the terms, write it outside
the bracket, then decide what is needed inside the
bracket.
multiplying two brackets: multiply each term
in the first bracket by the second bracket, expand
the brackets and simplify the resulting expression.
Alternatively use the grid method.
8.1 Expanding brackets
factorising the quadratic expression
x²  bx  c:
find a pair of numbers whose product is c and
sum is b
use these two numbers, p and q, to write down
the factorised from (x  p)(x  q)
using the difference of two squares: any
expression that can be written in the form a²  b²
can be factorised using the result
a²  b²  (a  b)(a  b).
C
1
Expand and simplify
a 5(z  1)  3z
b 4p  2(p  1)
c 4(h  1)  3h
d 2(d  3)  4(d  7)
e 6a  b  4(a  b)
f 4(5m  n)  5(n  m  3)
2
Expand and simplify
a 4(y  9)  2(y  7)
b 5(2a  1)  2(a  6)
c k  4(k  3)
d d(d  3)  2(d  1)
e 3n(n  4)  n(5n  1)
f 4t(2  5t)  3t(1  t)
3
Expand and simplify
a 4(t  4)  5(t  1)
b 4(h  3)  2(h  6)
c 3g(g  2)  g(g  2)
d 5e(2e  3)  e(4  e)
e 3s(s  5)  2(2  s)
f q(p  q)  p(p  q)
4
Expand and simplify
a 9s  5(s  1)
c 6f 2  3f(f  2)
e 2g  3g(g  h)
Exercise 8A
Questions in this chapter are targeted at the grades indicated.
D
C
1
2
3
Expand
a 3(x  4)
b 2(z  2)
c 3(m  n)
d 4(6  y)
e 3(x  2y  4)
f 4(2d  5)
g 3(x²  2)
h 2(g²  3g  2)
Expand
a p(p  3)
b q(q  2)
c 2x(x  4)
d h(3  h)
e c(b  a)
f s(6s  5)
g 2t(4t  2)
h 5x²(x  7)
Expand
a 2(b  4)
b 2(2x  7)
c m(m  3)
d 4y(5y  4)
e 4(h  2)
f 4g(2  g)
g 2z(z  4)
h 2n(4m  5n  6)
Exercise 8B
Watch Out!
You must multiply out the
brackets before you collect
like terms.
Check your signs.
b 12r  4(r  7)
d 8n  2n(n  1)
f 5p  2p(2  p)
Unit 2
8.4 Factorising quadratic expressions
8.2 Factorising by taking out common factors
h
i
j
k
l
B
Exercise 8C
D
C
1
2
Factorise
a 10x  5
b 2y  8
d 21g  7
e 10s  2t
g 45u  5v  20w
h ac  bc
i u  uv
k 4h²  3h
l 4q²  7q
n 3b  5b²
o q³  6q
Factorise completely
a 3xy  3xt
c 8pq  4ps
e 4ab  12ac  8ad
g 6x² 3x
i 6f ²  12f 3
k 4cd²  16c²d
m 8pqr  16prs
o 10x²y  25x²y²
b
d
f
h
j
l
n
p
c 2y  10z
f 6j  18k
A
j 6x²  8x  2
m 4y²  y
p a²  2a3
2ab  6ac
12xy  4y
mpq  mq
12t²  36t
3h4  h²
2a³b  2ab³
14a²b  8ab²  20ab
(2g)²  2g
3
1
Factorise
a (x  4)²  2(x  4)
b x(x  y)  y(x  y)
c z(z  4)  5z
d (5a  b)(3a  b)  3(3a  b)
e (a  9)²  2(a  9)
f (3g  2)²  2(3g  2)
2
Factorise completely
a 12(h  2)²  4(h  2)
b 25(x  3)² 10(x  3)
c 8(c  5)²  16(c  5)
d 3(q  4)  6(q  4)²
e 6(a  b)(a  b)  24(a  b)
f 8x²(x  2)  6x(x  2)
A
Expand and simplify
a (x  2)(x  4)
c (x  3)(x  5)
e (y  2)(y  1)
g (g  7)(g  3)
Expand and simplify
a (x  3)(2x  3)
c (2x  4)(x  3)
e (2z  3)(z  1)
g (3k  12)(2k  3)
i (3y  1)(4y  3)
k (3p  1)2
b
d
f
h
j
l
(x  1)(2x  1)
(y  2)(3y  2)
(2t  3)(3t  1)
(2a  3)(2a  4)
(2s  2)(3s  1)
(2d  3)2
Expand and simplify
a (x  3y)(x  2y)
c (x  3y)(x  2y)
e (2a  3b)(3a  2b)
g (4x  5y)2
b
d
f
h
(x  3y)(x  2y)
(x  3y)(x  2y)
(3m  2n)(2m  3n)
(4x  5y)2
1
Write down a pair of numbers:
a whose product is 12 and whose sum is 7
b whose product is 21 and whose sum is 10
c whose product is 15 and whose sum is 8
d whose product is 6 and whose sum is 1
e whose product is 12 and whose sum is 4
f whose product is 16 and whose sum is 0.
2
Factorise
a x2  5x  6
c x2  13x  42
e x2  9x  20
g x2  x  6
i x2  x  42
k x2  6x  112
m x2  100
b
d
f
h
j
l
x2  9x  20
x2  5x  6
x2  13x  42
x2  x  20
x2  4x  5
x2  25
Exercise 8G
Exercise 8E
1
Note that (a  b)2 is not equal
to a2  b2.
Exercise 8F
8.3 Expanding the product of two brackets
B
Watch Out!
8.4 Factorising quadratic expressions
Exercise 8D
B
2
(x  3)2
(d  6)2
(p  5)2
(x  y)2
(p  q)2
Examiner’s Tip
A
b (x  3)(x  2)
d (y  3)(y  2)
f (z  4)(z  6)
1
Factorise
It will help you in the
a x2  81
examination if you learn
b x2  64
a2  b2  (a  b)(a  b).
c y2  121
d 49  y2
e z2  900 f 225  z2
2
g (x  1)  5
h 64  (8  y)2
i (c  d)2  (c  d)2
57
58
Chapter 8 Expanding brackets and factorising
A
2
A02
3
Without using a calculator, use algebra to find
the value of:
a 642  492
b 4.52  3.52
2
2
c 0.375  0.125
d 9052  8952
Unit 2
Exercise 8H
A
Factorise these expressions, simplifying your
answers where possible.
a 9x2  100
b 4a2  1
c 8t2  144
d 16  (z  2)2
2
2
e (2g  1)  (2g  1)
f (a  b  1)2  (a  b  1)2
g 36(a  _12 )2  144(b  _12 )2
2
h 16(m  n)2  16(m  n)2
4
1
Factorise completely
a 3x2  27
b 5h2  20
c 40r2  9000
d 4p2  36q2
2
2
e 12u  48v
f 3(d  1)2  3(d  1)2
A
3
Factorise
a 5x2  8x  3
c 3x2  7x  4
e 6x2  10x  5
g 5x2  23x  12
i 8y2  10y  3
k 7y2  27y  4
m 4z2  8z  5
o 6z2  22z  12
b
d
f
h
j
l
n
Factorise completely
a 6x2  13x  5
c 5x2  7x  6
b 6y2  25y  4
Factorise
a x2  2xy  3y2
c 6x2  3xy  3y2
b 2x2  11xy  5y2
2x2  7x  5
6x2  9x  6
6x2  8x  3
4x2  7x  3
2y2  13y  21
3y2  3y  6
6z2  7z  3