Download 0407AlgebraicExpress..

Algebraic
Expressions
Polynomials
Operation of Powers
and
Exponent Properties
the nth power of a or
a  a  a2
a  a  a  a3


a to the power of n
index
a  a   a  a n
base
n a's
Rules :
1.
a m  a n  a m n
For example 23  22  23 2
 25
 32
3
2
3 2
For example m  m  m
 m5
Simplify the following expressions.
eg
eg
eg
x x 
3x 5x
6 x 3x 
4
6
4
2
3
eg
3x  7 x 
2x  4x 
 4x  4x 
eg
 x3  4x3
eg
 x3  3x 2
eg
 6 x5  2 x 4
eg
eg
eg
eg
eg
2
4
5
3
2
4
x y x y 
x y xy 
2x y  3xy 
2
3
2 2
5
3 2
4
Chargellenge questions
eg
2 3
5 n
m 6
It is given that a b  a b  a b .
Find the value(s) of n and m.
eg
n
It is given that x  9 and
x m  81. Find the value of x m n .
2.
a 
m n
 a mn
 
For example 2
3 2
 232
 26
 64
For example a 
3 4
 a 34
 a12
Simplify the following expressions.
eg
eg
eg
eg
eg
eg
3.
x 
 x 
 x 
 x 
2x 
3 4
2 9
3 4
2 6
4 4
 3x 
2 3
abn  a nb n
2
2
2
For example 2  3  2  3
 49
 36
 
4
34 14
For example a b  a b
3
 a12b 4
Simplify the following expressions.
eg
eg
eg
eg
eg
eg
3 2 5
2 3 2
3 2 4
 3x y 
2
3
eg
Find the 5th power of
x.
eg
Find the 5th power of
x2 .
eg
4.
xy 
2 3 4
x y 
x y 
6 x y 
 2 x y 
3 2
Find the 5th power of  x  .
a n  a m  a nm
For example 24  23  243
2
For example a6  a 4  a6 4
 a2
Simplify the following expressions.
eg
x3  x 4
eg
15x3  5 x
eg
24 x5   6 x 2


eg
 45x6  5 x7
eg
 128x5   16 x3
eg
 256x5  16 x


eg
x4 y6
xy 5
32 x5 y 6
8 xy 2
 25x 4 y 2
5 xy 6
3x 2  9 x 4  2 x
eg
4 x  8x 2 2 x3
eg
36 x12  2 x 7  3x 2
eg
eg
eg
  
4 x 3x
5
eg
x8
2 x  3x 
3
eg
eg
eg
2
 8x
4 x 2  12 x 7
 
 3x 3
 
2 x 2 y 3x 2
6 xy 2
2 x y  xy
2
2
eg
eg
x
4 xy 3
2
x y  6 x 
2
3

x3 y m a
eg If n 7  2 , where m and n are
x y
b
positive integer. Find the values of m
and n.
Some common mistakes are always happened.
1.
m2 n6  mn8 X
2.
3  42  122 X
2
2
2


a

b

a

b
3.
X
a
 ab X
4.
b
Conclusion
1. a  a  a
m
n
m n
2.
a 
3.
abn  a nb n
4.
a n  a m  a nm
m n
 a mn
HW p7.40 #2(b)(d), 4(a)(c) and 5(b)(d)
An algebraic expression is made up of the
signs and symbols of algebra. These
symbols include the Arabic numerals, literal
numbers, the signs of operation, and so
forth. Such an expression represents one
number or one quantity.
Thus, just as the sum of 4 and 2 is one
quantity, that is, 6, the sum of c and d is one
quantity, that is,
ab , a  b ,
cd .
Likewise
a
b,
b,
and so forth, are algebraic
expressions each of which represents one
quantity or number.
The terms of an algebraic expression are
the parts of the expression that are
connected by plus and minus signs. In the
expression 3abx  cy  k , for example,
3abx , cy , and k are the terms of the
expression.
The following expressions are the examples
of polynomial.
Class Practice
1.
x
2.
3y  a  b
3.
abx
4.
5.
6.
4  2b  y  z
3y2  4
2y
1
6
1. Monomial
2. Trinomial (also polynomial)
3. Monomial
4. Polynomial
5. Binomial (also polynomial)
6. Binomial (also polynomial)
In general, a COEFFICIENT of a term is any
factor or group of factors of a term by which
the remainder of the term is to be multiplied.
Thus in the term
2axy , 2ax is the
coefficient of y 2ax y  . 2a is the
coefficient of xy 2a xy  , and 2 is the
coefficient of axy 2 axy  .
The word "coefficient" is usually used in
reference to that factor which is expressed
in Arabic numerals. This factor is sometimes
called the NUMERICAL COEFFICIENT. The
numerical coefficient is customarily written
as the first factor of the term. In 4 x , 4 is the
numerical
coefficient,
coefficient, of
x.
or
simply
the
2
Likewise, in 24xy , 24 is
2
the coefficient of xy and in 16a  b , 16 is
the
coefficient
numerical
of
a  b 
coefficient
is
.
When
written
no
it
is
understood to be 1. Thus in the term
xy ,
the coefficient of xy is 1.
Grouping like terms or COMBINING like
terms
eg
eg
7 x  5x  2 x
2
2
2
2
5b x  3ay  8b x  10ay
2
2
2
step 1. indicate like terms by using pen
or pencil with different indicators such
as ﹏﹏ or
or others.
5b x  3ay  8b x  10ay
2
2
2
2
 5b 2 x  8b 2 x  10ay 2  3ay 2
step 2.
grouping LIKE terms.
 3b x  7ay
2
2
step 3. combining LIKE terms.
Simplify the following expressions.
1.
2.
3.
4.
5.
2a  4a
y  y2  2 y
4ay ay

c
c
2ay 2  ay 2
bx 2  2bx 2
6.
2 y  y2
Answer
1. 6a
2. y 2  3 y
3.
3ay
c
4. ay 2
5. 3bx 2
6. 2 y  y 2
Dissimilar or unlike terms in an algebraic
expression cannot be combined.
For example,  5 x  3xy  8 y
2
2
contains
three dissimilar terms.
This expression cannot be further simplified
by combining terms through addition or
subtraction.
The
rearranged
as
y3x  8 y   5 x 2
expression
may
x3 y  5 x   8 y 2
,
but
such
be
or
a
rearrangement
is
not
actually
a
simplification.
OPERATIONS WITH POLYNOMIALS
Arranging a polynomial in order(descending
order)
Adding and subtracting polynomials is
simply the adding and subtracting of their
like terms. There is a great similarity
between the operations with polynomials
and denominate numbers. Compare the
following examples:
Addition and subtraction of polynomial
1. Add
3qt  2 pt
to
5qt  pt .
Method I
3qt  2 pt  5qt  pt
 3qt  5qt  2 pt  pt
 8qt  3 pt
Method II
3qt  2 pt
 ) 5qt  pt
8qt  3 pt
2. Add
5x  y
to
3x  2 y .
Method I
5 x  y  3x  2 y
 5 x  3x  y  2 y
 8x  3 y
Method II
5x  y
) 3x  2 y
8x  3 y
3. Add
x 3  3x  2
to
4 x 4  3x 3  3x .
Method I
x 3  3x  2  4 x 4  3x 3  3x
4
3
3
 4 x  x  3x  3x  3x  2
 4 x 4  2 x3  6 x  2
Method II
4 x 4  3x 3  3x
)
x 3  3x  2
4 x 4  2 x3  6 x  2
One method of adding polynomials (shown
in the above examples) is to place like terms
in columns and to find the algebraic sum of
the like terms. For example, to add
3a  b  3c , 3b  c  d ,
and
2a  4 d ,
we would arrange the polynomials as
follows:
3a  b  3c
)
2a
3b  c  d
 4d
5a  4b  2c  3d
Subtraction may be performed by using the
same arrangement-that is, by placing terms
of the subtrahend under the like terms of the
minuend and carrying out the subtraction
with due regard for sign. Remember, in
subtraction the signs of all the terms of the
subtrahend must first be mentally changed
and then the process completed as in
addition. For example, subtract 10a  b
from 8a  2b , as follows:
8a  2b
) 10a  b
 2a  3b
Again, note the similarity between this type
of
subtraction
and
the
subtraction
of
denominate numbers.
Addition and subtraction of polynomials also
can be indicated with the aid of symbols of
grouping. The rule regarding changes of
sign when removing parentheses preceded
by a minus sign automatically takes care of
subtraction.
For example, to subtract
10a  b
8a  2b
the
,
we
can
use
from
following
arrangement:
8a  2b   10a  b   8a  2b  10a  b
 2a  3b
Similarly, to add  3 x  2 y  to  4 x  5 y  ,
we can write
 3x  2 y    4 x  5 y   3x  2 y  4 x  5 y
 7 x  3 y
Practice problems. Add as indicated, in each
of the following problems:
Find the sum of the following
3a  b
1.
 ) 2a  5b
2.
3s t  3s t  st  5  4s t  5
3
2
3
4a  b  c   a  c  d   3a  2b  2c 
3.
4x  2 y
3x  y  z
) 3x
z
4.
In problems 5 through 8, perform the
indicated
operations
terms.
5.
6.
7.
2a  b   3a  5b 
5x y  3x y   x y
3
2
3
x  6  3x  7

 
8. 4a  b  2a  b
2
2
Answers:
1.
5a  4b
2.
7 s 3t  3s 2t  st
3.
8a  3b  4c  d

and
combine
like
4.
8x  y
5.
 a  4b
6.
4 x 3 y  3x 2 y
7.
 2x 1
8.
2a 2  2b
Class Work p7.28 #1(a), (b), (e), (f), 2
HW. p7.30 #12, 14(a) (d) 19, 20
MULTIPLICATION OF A POLYNOMIAL BY
A MONOMIAL
We can explain the multiplication of a
polynomial by a monomial by using an
arithmetic example. Let it be required to
multiply the binomial expression, 7  2 , by
4. We may write this 4 7  2 or simply
47  2 . Now 7  2  5 . Therefore,
47  2   45  20 . Now, let us then
subtract.
47  2  4  7   4  2  20
Thus,
.
Both
methods give the same result. The second
method makes use of the distributive law of
multiplication.
When
there
are
literal
parts
in
the
expression to be multiplied, the first method
cannot be used and the distributive method
must be employed. This is illustrated in the
following examples:
45  a   20  4a
3a  b   3a  3b
ab x  y  z   abx  aby  abz
Thus, to multiply a polynomial by a
monomial,
multiply
each
term
of
polynomial by the monomial.
Practice problems. Multiply as indicated:
Expand the following expressions.
1.
2aa  b 
the
2.
3.
4.
4a 2 a 2  5a  2
 4 x y  3z 
2a 3 a 2  ab 
Ans
1.
2.
3.
4.
2a 2  2ab
4a 4  20a3  8a 2
4 xy  12 xz
2a5  2a 4b
Class Work p7.35 #1(a), (b), (d), (e), (h) 2
HW. p7.36 #10(a), 11(a) , 13(a), 15, 17