Math (4)_text.pdf

(3049 KB) Pobierz
x
y
x
—
ALGEBRAIC
EXPRESSIONS
AND
ALGEBRAIC
FORMULAS
I
)ét
rnc
:
t
hé
folio
vri
ni»
ternis
instance.
A
lût*
braie
Expressions
5x
-3jcH
2
and
When
we
V*
3xyn
operations
of
addition
and
subtraction
are
applied
to
algebraic
terms
x
—
(x*0)
are
al
gebraic
expressions.
obtain
an
algebraic
expression.
For
DSEBEBI9
A
polynomial
Where
n
in
the
variable
x
is
an
algebraic
expression
of
the
form
1
P(x)=a
n
x
n
+a
n_
x
n
l
~
l
+a
n
_
2
x”~
2
+...+a
x+a„,a„
power
of
is
t
the
highest
x,
is
a
non-negative
integer
cailed
the
degree
of
the
polynomial
and
each
coefficient
a„
of
x
is
a
real
ntimber.
The
2
coefficient
a„
of
the
highest
power
cailed
the
leading
coefficient
of
the
polynomial.
7.
4
3
y
+
2
y
+8x
2
is
a
polynomial
in
two
variables
x
and
y
and
has
degree
Katiomil
K'vpress
K’xprvssifiri
The
quotient
Pix)
q(x)
of
two
Reduce
the
following
algebraic
tractions
to
their
lowest
forms.
polynomials
p(x)
and q(x
)
is
,
where
q(x)
is
Ix+mx-ly—my
(i)
a
non-zero
polynomial,
cailed
a
3jc
2
3x
-3y
2
2
(ii)
rational
expression.
+18jc+27
For
example,
,
3x+8*0
3x+8
rational
expression.
isa
Solution
2
5x
—
45
In the
rational
expression
Pix)
q{x)
bc+mx-ly-my
(i)
2
3x
-3
2
p(x)
is
cailed the
numerator
and
q(x)
is
x(l
+m)—y(l+m)
3(x
2
known
as
the
denominator
of
the
rational
-
y
2
)
P(x)
expression
q{x)
The
rational
expression
(l+m)(x-y)
3(x+y)(x-y)
Pix)
q(x)
need
not
be
a
polynomial.
l+m
Which
2
is
in
the
lowest
tbrms.
3(
x
+
3)
(ü)
3x
+
18
x+
27
_
3(x
2
+6x+9)
2
2
5x
—45
5(x
-9)
5(x-3)
Which
is
in
the
lowest
fomis
3(x+3)(x+3)
5(x+3)(x-3)
Simpiify
2x
(i)
x-y
x
+y
2x
(«)
l
j
r-y
+-
1
2
x
2
je
x
-16
-Solution
1
—
4
x+2
1
2x
2
je
(i)
*-y
x+y
1
1
-y
2x
x-y x+y
(x+y)(x-y)
x+y-(x-y)+2x
(jc+yXJf-y)
(L.C.M
of
denominators)
/
+
y-,/+y+2x
(x+y)(x-y)
2x+2y
(x+y)(x-y)
2(x+y)
(•*+?)(*-
y)
=
J
2
x-y
x+2
-+-
1
2x
(«)
c
2
4
2
-16
2
x
-4
2x
(x
2
x
+4
)(x
2
-4)
2x
2
x
2
—4
x+2
x
(x+2)(x-2)
C*
2
+
4)(
x
+
2)
(x-
2)
x+2
-
té
+/+
tfc
+
4)(x
+
2)
(x
-
2)
2
2
_
2x
-x(x
+4)+(x
+4)(x-2)
(x
1
^
f
(x
+
4)(x
+
2)
(x-
2)
2
y
5
-8
{x
2
+4){x+2){x-2)
-8
-8
2
(x
+4)(x
-4)
2
4
a:
-16
Ixy
\
4y
Find
the
product
x+2
2x-3
y
4x
~9y
xy
+
2y
2
2
x
~4x+4
Ixy
x
-4
x
-4x+4
Solution
2
4x
-9
2
2
_
(x+2)[(2x)
-(3
y)
)
xy+2y
(2x-3y)(x+2)y
2
x
x
2
-4
V
14y
x+2
2x-3
y
2
(x+2)(^ï)
Ixy
x-
(x-2
)(^f)
x(x+2)
2(x—2)
14
y
^^x+^yX^Sy)
y{yXÎ){2^Sy)
2x+3y
3
*
Evaluate
jf
x
=
—
4
and
y=9
5
(x+y)
Solution
We
hâve,
by
putting
x
=
-4
and
y
~
Ixy
Simplify
t
\4y
9,
2
—
4x+4
x
2
-4
3x
2
Jÿ+6
__
3
(-
4
)
\[9+6
5
(-
4
+
9
)
2
3
(
16
)(
3)+6
5
(
5
)
150
=6
25
5{x+y)
Solution
Exercise
4.1
1.
Identify
whether
the
following
2.
State
whether
each
of
the
following
algebraic
expression
are
expression
is
a
rational
expression
polynomials
(yes
or
no).
(i)
or
not.
3x
2
+—
—
X
No
No
Yes
3-v/x
(i)
No
3Vx+5
2
(H)
(ni)
(iv)
3x
2
3
-4x
-xVx
+3
x
(«)
3
-2x
2
+
V3
x
-3x
+
>/2
3X
2x
-1
+
8
No
(Hi)
2+3x-x
2
2
x
+
6x+9
2
^
x
—9
Yes
Yes
2,-yfx
+3
foJlowing
<iv)
No
rational
2Vx-3
Reduce
2
(x-2y
2(x
-
2)(x
+
2)
the
>^ZT(x-
2)
(x
expression
to
the
lowest
forms.
(i)
+
2)
120x
y
z
30x
y
3
2
3
s
=
4x
2
'
3
31
y
z
3
5
'
2
2(x
5
+
2)
=
_
4x"’
y
z
2
64x
-64x
(vit)
4yV
x
(8x
2
+8)(2x
+
2)
4
64x(x
-1)
8a(x+l)
(H)
4aOKf)
(x-l)(>-pf)
4a
8(x~+l).2(x
+
l)
2(x
(x
(iü)
2
-1)
x-1
~
4
64x(x
-1)
2
2
2
+
y)
-4xy
_
x
+
y
+
2xy-4xy
2
(x-y)(x-y)
(x-y)
2
2
_
x
+
y
-
2xy
J6(x
2
+1)(x
+
1)
2
2
4x(x
+
l)(x
-1)
(x
2
+l)(x
+
l)
(x-y)(x-y)
(x-y)
2
(x
4x^<îr(x-i)^tT
=
4x(x-l)
2
2
9x
-(x -4)
2
-
y)(x
-
y)
2
(3x)
2
2
-(x
~4)
(x
(iv)
3
-
y
)(x
-2xy
+
y
2
3
2
2
(x~y)(x
+
xy
+
y
2
)
4+3x-x
2
f
_(3x
+
x
2
-4)£x^x
+4)
4+3x-x
2
(^-k35T-x
2
)
(jè^y^ix-y)
2
=(x-yr
„
=
3x
+
x
2
-4
=
x
2
+
3x
-
4
x
—
y-2z.
xz
3
(x
(v)
2
+
2)(x
-1)
(x
+
l)(x
2
-4)
jisrtfix-iyÿisrf
4
.
Evaluate
(a)
.
,
x
for
(i)
x
=
_
j
x-i
y
=
-1, z
=
-2.
3
j>4<y(x-2)^K2y
x
(vi)
x-2
(3)
(-
1)
-
2(-2)
(a)
2
-4x+4
(
x
-
2)
2
3(—
2)
_
-27
+
4
"
-6
2x
2
-8
2
2(x
-4)
_-23_23
”
-6
6
5
6
5
x v
3
-5z
4
2
(b)
for
x
=
xyz
4,y
=
-2,z
=
_
(x-5)(x+5)-(x
+
5)(x-6)
(x
+
6)(x-6)
-1
_
(x+5)[(x-5)-(x-6)]
2
3
_
(4)
(—
2)
—
(—
I)
4
_
-
1
6(8)
-
5
8
(x+6)(x-6)
(4)(—2)(—
1)
_
(x+5)(/->^
+
6)
(x
_
-128-
5
8
5.
-133
8
"
5
6
8
(x
+
6) (x
-
6)
Perform
the
indicated
operation
and
simplify
15
_
x+5
(x+6)(x-6)
x
2
-36
(iv)
+
5)(1)
G)
4_
y
2xy
2
2
x
-
y
3y-2x
_
1
5(3y
-
2x)
-
4
(2x
-
3y)
(2x-3y)(3y-2x)
2x-3y
x-y
x
+
y
_
x(x
+
y)
-
y(x
-
y)
(x-y)(x
+
y)
2x$
2
x
-
y
2
_
45y-30x-8x+12y
(2x-3y)(3y-2x)
y>
_x
2
+^jT-^
+
y
2
x
+
y
2
2
2xy
_
57y—
38x
(2x-3y)(3y-2x)
2xy
19^^xT
(H)
(2x
-
3y)^~2xf
~
19
x
2
2x
-
3y
+
y
-2xy
2
2
(x
-
y
)
(x-y)"
2
(x
2
+
2x
l-2x
l
1
-
2x
+
2x
l
_
(1
+
2x)
2
-
(1
—
2x)
2
(l-2x)(l+2x)
+
y)
_
x-y
X
+
y
_
(l
+
4x
+
4x)-(l+4x
—
4x)
(l-2x)(l
+
2x)
2
2
x
2
+6x
+
9
2
2x
-18
x-2
x
2
x
2(x
+
2
2
+3x+3x
+
9
-9)
x
+
2
_
j
+>*^+4
xyi
p>45^+4x
(l-2x)(l
+
2x)
8x
(1—
2x)(l
+
2x)
x
(ni)
2
x-2
x(x+3)
+
3(x
+
3)
2(x-3)(x
+
3)
x
+
2
8x
X-2
2
l-4x
_
(x+3)(x+3)
2(x-3)(x
+
3)
-25
-36
x
x
+
5
x
2
+
6
_
2(x-3)(x-2)-(x+3)(x
+
2)
2(x
-
3)
(x
+
3)
(x
+
3)
_
(x-5)(x+5)
(x-6)(x
+
6)
x+5
x
+
6
_
2(x
2
-2x-3x
+
6)-(x
2
+
2x
+3
x
+
6)
2(x-3)(x
+
3)
2

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin