Download MATH0111 ANSWERS I 1. (a) (x - 5y)(3x - 2y 2)=(x - 5y)(3x)+(x

MATH0111
ANSWERS I
1. (a) (x − 5y)(3x − 2y 2 ) = (x − 5y)(3x) + (x − 5y)(−2y 2 )
= 3x2 − 15xy − 2xy 2 + 10y 3 .
(b) We have
(z − y)3 = (z − y)(z − y)2 = (z − y)(z 2 − 2zy + y 2 )
= (z − y)z 2 + (z − y)(−2zy) + (z − y)y 2
= z 3 − yz 2 − 2yz 2 + 2y 2 z + y 2 z − y 3 = −y 3 + 3y 2 z − 3yz 2 + z 3 .
(c) (y 2 −3z)(y 2 +3z) = (y 2 −3z)(y 2 )+(y 2 −3z)(3z) = y 4 −3y 2 z+3y 2 z−9z 2 =
y 4 − 9z 2 . You can see this more quickly as the “difference of two squares”.
2. (a)
(b)
2
3(a + 4) + 2(a − 2)
5a + 8
3
+
=
=
.
a−2 a+4
(a − 2)(a + 4)
(a − 2)(a + 4)
y
x
y(y − 3) + x(x + 1)
x2 + x + y 2 − 3y
+
=
=
.
x+1 y−3
(x + 1)(y − 3)
(x + 1)(y − 3)
The technique in the next few questions is the same, so I’ll just do one in
detail.
3. (a) x2 + 8x + 7. The idea here is to find two numbers, say a and b,
such that a + b = 8 and a.b = 7. Clearly we can take a = 7, b = 1, so
x2 + 8x + 7 = (x + 7)(x + 1). Check by multiplying out.
(b) x2 − 12x + 20 = (x − 10)(x − 2).
(c) z 2 − 49 = (z − 7)(z + 7)
(difference of two squares).
(d) y 2 + y − 30 = (y + 6)(y − 5).
(e) x4 − 25x2 = x2 (x2 − 25) = x2 (x − 5)(x + 5).
(f) 20+9x+x2 = (4+x)(5+x). If you are happier writing it as x2 +9x+20 =
(x + 4)(x + 5), that is equally good.
(g) 2x3 − 8x2 − 24x = 2x(x2 − 4x − 12) = 2x(x − 6)(x + 2).
(h) 3y 2 − 75 = 3(y 2 − 25) = 3(y − 5)(y + 5).
4. (a) x2 − 8x + 15 = 0 when (x − 5)(x − 3) = 0. So either x − 5 = 0 or
x − 3 = 0. Thus x = 5 or 3.
(b) x2 − 12x + 36 = 0 when (x − 6)2 = 0. Thus x = 6 (a repeated root).
(c) a2 + 5a = 6 when a2 + 5a − 6 = 0. So (a + 6)(a − 1) = 0. Thus a = −6
or 1.
(d) z 2 + 14z + 24 = 0 when (z + 12)(z + 2) = 0. Thus z = −2 or −12.
(e) x3 − 4x2 + 4x = 0 when x(x2 − 4x + 4) = 0. So x(x − 2)2 = 0. Thus
x = 0 or 2.
(f) 2z 2 − 16z + 32 = 0 when 2(z 2 − 8z + 16) = 0. So 2(z − 4)2 = 0. Thus
z = 4 (only).
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