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Chapter R- Review
R.2 Integer Exponents
Definition
(i) for any real number a and a positive integer n (1,2,3,4….) a n  a
a

a



n times
3
Example: (2) = 222= 8
(ii) for any nonzero real number a: a0 = 1
Example: (-312)0= 1
1
an
Remark: a in the above definition is called the base of the exponent and n is called an exponent or a power
Example: (3) 2  1 2  1
(3)
9
2
2
Caution: -3 = -9 but (-3) = 9
(iii) for any nonzero real number a and a positive integer n: a n 
a n  a m  a n m
an
1
 a nm  mn
m
a
a
n m
nm
(a )  a
Laws of exponents (must be memorized)
1
2 5  2 4  25 4  2 1 
2
7
2
 2 7 5  2 2
5
2
(3) 
4 2
 (3) 42  (3)8
(2 x ) 5  25  x 5
( a  b) n  a n  b n
2
n
22
4
2
   2  2
x
x
 x
an
a
   n
b
b
Note that these properties can also be applied in reverse
x 5  x 23  x 2  x 3
 
x 9  x 33  x 3
3
81x 4  34  x 4  (3x) 4
Additional properties of exponents:
n
a
b
   
b
a
n
m
a
b
 n
m
b
a
n
2
2
32 9
2
3
     2 
2
4
3
2
x 3 y 5

y 5 x 3
Caution: Identify the base of the exponent properly. Example:
2
) 1
Example: Simplify the following expression 4 x 3 ( yz
4
2 x y
4 x 2 ( yz ) 1
4
4 1
1


 6 2
3 4
2
4
1
2
2 4
2 x y
x 8 x y( yz )
8 x yyz 2 x y z
1
1
x2


2
2
2x
2  ( x)
2
R.4 Polynomials in one variable
A monomial: an algebraic expression of the form axn, where a is a real number, x is a variable and n is a nonnegative
integer. Example: 2x3,
3x 5 , 7
A binomial is the sum (or difference) of two monomials. Example: 2x3 -
3x 5
A trinomial is the sum (and/or difference) of three monomials. Example: 2x3 - 3 x5 + 7
A polynomial is the sum and/or difference of many monomials. It can be written in general as
anxn + an-1xn-1 + an-2 xn-2 + … + a1x + a0
The numbers an, an-1, …, a0 are called the coefficients of the polynomial. The highest power of x is called the degree of
the polynomial and the coefficient of the monomial with the highest degree is called the leading coefficient.
We say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x
Example: - 2x3 - 4x + 5 is a polynomial of degree 3 and the leading coefficient is -2.
3x  1
,
x 1
x  1 are not polynomials (not sums of monomials)
The like terms are monomials with the same variable raised to the same power. Example: 2x5 and
but 2x3 and
3x 5 are like terms
3x 5 are not.
Operations on polynomials:
Addition/subtractions
(a) Use the distributive property to eliminate any parentheses
(b) Identify and add/subtract the like terms (add/subtract the coefficients of the like terms)
(c) Write the polynomial in the standard form
Example
3(2 x 3  4 x 2  x  3)  (4 x 2  2 x  5) 
6 x 3  12 x 2  3x  9  4 x 2  2 x  5 
6 x 3  (12 x 2  4 x 2 )  (3x  2 x)  (9  5) 
6 x 3  (12  4) x 2  (3  2) x  14 
6 x 3  8 x 2  5 x  14
Multiplication
To multiply two monomials with the same variable, multiply their coefficients and raise the variable to the sum of their
exponents. Example: 2x2(-3x3 )= (2)(-3)x2+3 = -6x5
Two multiply two polynomials
a) multiply each term of one polynomial by each term of the other polynomial
b) identify and combine like terms
c) Write the polynomial in the standard form
Example
(2x2 +3)(-3x3 + 4x – 5) = (2x2)(-3x3)+ (2x2)(4x)+(2x2)(-5) + (3)(-3x3)+(3)(4x)+(3)(-5)=
-6x5
+ 8x3
-10x2
-9x3
+ 12 x - 15 =
5
3
2
-6x – x -10x + 12x -15
Special multiplication formulas (must be memorized)
(*)
(a - b)(a + b) = a2 - b2
(**)
(a + b)2 = a2 + 2ab + b2
(***)
(a - b)2 = a2 – 2ab + a2
Example
a) (2x-1)(2x+1) = (2x)2 – 12 = 4x2 – 1
b) (x+3)2 = x2 + 23x + 32 = x2 + 6x + 9
Long division of two polynomials
To divide two monomials with the same variable, divide the coefficients and subtract the exponents of the variable.
Example:
6 x 3 6 31
 x  2x2
3x 3
Example: Divide 6x3 + 7x2 + 3 by 3x – 1
Remark: 6x3 + 7x2 + 3 is called the dividend and 3x-1 is called the divisor. You can perform division only when the degree
of the dividend is greater or equal to the degree of the divisor.
1. Write the problem as a long division problem. Write both polynomials in the standard form. Include the missing
terms, if any, with the coefficient zero
_2x2 + 4x + 3_____ this is the quotient
3x -1  6x3+ 10x2 + 5 x + 3
divide 6x3 by 3x
6x3 – 2x2
multiply 2x2(3x-1) and subtract;
2
12x + 5x + 3
divide 12x2 by 3x
2
12x – 4x
multiply 4x(3x-1) and subtract;
9x + 3
divide 9x by 3x
9x - 3
multiply 3(3x-1) and subtract
6
this is the remainder
2. Divide the first term of the dividend (6x3) by the first term of the divisor (3x)
6x3
= 2x2
3x
and write it above the line
3. Multiply 2x2 by the divisor (3x-1) and write it beneath the dividend
2x2 (3x-1) = 6x3 – 2x2
4. Subtract: 6x3 + 10x2+ 5x + 3 – (6x3 -2x2) = 6x3 + 10x2 +5x +3 -6x3+ 2x2 = 12x2+ 5x + 3
5. Repeat the steps 2 – 4 with 12x2+ 5x + 3 as the dividend
6. Divide the first term (12x2 ) by the first term of the divisor (3x)
12 x 2
 4x
3x
And write it above the line
7. Multiply 4x by the divisor (3x-1): 4x(3x-1) = 12x2 – 4x
8. Subtract ( 12x2 + 5x + 3) – (12x2 - 4x)= 12x2 +5x +3 -12x2 + 4x = 9x + 3
9. Repeat steps 2- 4 with 9x+3 as the dividend
9x
3
3x
Multiply 3(3x-1) = 9x - 3
Subtract (9x + 3) – (9x-3) = 9x +3 -9x + 3 = 6
10. STOP when the degree of the polynomial that plays the role of the dividend is less than the degree of the
divisor
Note that
dividend = (quotient)(divisor) + remainder
or
dividend
remainder
 quotient 
divisor
divisor
In the example above, we can write
6x3+10x+5x+3 = (3x-1)( 2x2+ 4x + 3)+ 6
Or
6 x 3  10 x 2  5 x  3
6
 2x2  4x  3 
3x  1
3x  1
R.5 Factoring polynomials
In the product 3(2x-1)(3x2 +x + 2), the polynomials 3, (2x-1), and (3x2 + x+ 2) are factors.
To factor a polynomial means to write it as a product of two or more polynomials
How to factor a polynomial:
(i)
Factor out common factors:
- Identify common factor (if any) in all terms of the polynomial
- Use the distributive property: ca + cb = c(a+ b) to write the polynomial as a product
Example: a) 6x3 + 2x2 = 23x2x + 2x2 = 2x2 (3x + 1)
b) (x+3)(x-2) + (x-2)x = (x+3)(x-2) + (x-2)x = (x-2)[(x+3) +x] =(x-2)(2x+3)
c) 5(2x+1)2 + (5x-6)2(2x+1)2= 5(2x+1)2 + (5x-6)2(2x+1)2
= (2x+1)[5(2x+1)+ 4(5x-6)]
= (2x+1)[10x+5 + 20x – 24] =(2x+1)(30x-19)
(ii)
Use a formula
- a2 – b2 = (a-b) (a+b)
- a3 - b3 = (a-b) (a2 + ab + b2)
- a3 + b 3 = (a+b)(a2 – ab + b2)
Example: a) 4x2 – 25 = (2x)2 – (5)2 = (2x - 5)(2x + 5)
b) x3 + 8 = (x)3 + (2)3 = (x+2)(x2 – 2x + 4)
c) 27- 64x3 = (3)3 –(4x)3 = (3 - 4x)(32 + 3(4x)+ (4x)2)= (3 – 4x)(9 + 12x+16x2)
(iii)
factor by grouping
- group the polynomial into two (or more groups)
- factor each group
- if there is a common factor for each group, factor it out and factor the remaining polynomial.
If there is no common factor for all the groups, group the original polynomial differently and try again.
Example:
a) 3x2 + 6x –x – 2= (3x2 + 6x) + (-x – 2)= 3x (x +2) + (-1)(x+2) = (x+2)(3x – 1)
b) x4 + x3+ x + 1= (x4 + x3) + (x+1) = x3(x+1) +(x+1)= (x+1)(x3+ 1)
= (x+1) )(x3 + 13) =(x+1)(x+1)(x2 – x+1) = (x+1)2(x2- x +1)
(iv)
Factor a trinomial ax2 + bx + c, where a, b, c are integers
(iva) Factoring a trinomial x2 + bx + c
- Find two numbers p, q such that
pq = c and p + q = b
- If such two numbers exist, then
x2 + bx + c = (x + p)(x + q)
Remarks: a) if c is negative, then p and q have opposite signs;
b) if c is positive then p, q have the same signs and if b is negative,
then p,q are negative and if b is positive then p,q are positive
Example: Factor x2 + 5x + 6.
We look for two numbers p and q such that pq = 6 and p + q = 5. Those numbers are 2
and 3. Therefore, x2 + 5x + 6 = (x+2)(x+3)
(ivb) Factoring a trinomial ax2 + bx + c, a  1, a,b,c have no common factors
- Find two numbers p and q so that
pq = ac and p + q = b
- Rewrite the trinomial as
ax2 + bx + c = ax2 + px + qx+ c
- Factor by grouping
Example: Factor -6z2 + z + 1
We look for two numbers p, q such that pq = -61 and p + q = 1.
Those numbers are -2 and 3.
Therefore
-6z2 + z +1 = -6z2 - 2z + 3z + 1= (-6z2 - 2z ) + (3z + 1)=
-2z(3z + 1) + (3z + 1) = (3z +1)(-2z + 1)
(v)
Combine the above methods
Remark : - If a polynomial cannot be factored we say it is prime
- A polynomial must be factored completely, which means that each factor must be prime
R.7 Rational Expressions
3
2
A rational expression is a quotient of two polynomials. Example 2 x 2 4 x  x  5
2 x  3x  6
Reducing a rational expression to the lowest terms
A rational expression is reduced to the lowest terms if the numerator and the denominator have no common factors
To reduce a rational expression to the lowest terms:
- Factor completely the numerator and the denominator
-
Cancel common factors ( use the property: c  a  a )
c b
b
3x 2  x  2
x 1
Example:
=

2
x 1
3x  5 x  2
Multiplying and Dividing Rational Expressions
To multiply two rational expressions
- multiply the numerators and the denominators a  c  a  c
b d
bd
- reduce to the lowest terms
Example:
x2  x  6
x 2  25
( x 2  x  6)( x 2  25)



x 2  4 x  5 x 2  2 x  15 ( x 2  4 x  5)( x 2  2 x  15)
( x  3)( x  2)( x  5)( x  5) x  2

( x  5)( x  1)( x  5)( x  3) x  1
To divide two rational expressions
-
multiply first rational expression by the reciprocal of the second  a  c  a  d 
b
reduce to the lowest terms
d
b c
Example:
x2  7x  6
2
2
2
2
2
2
x 2  x  6  x  7 x  6  x  5 x  6  x  7 x  6  x  5 x  6  ( x  7 x  6)( x  5 x  6) 
x 2  5x  6
x 2  x  6 x 2  5x  6
x 2  x  6 x 2  5x  6
( x 2  x  6)( x 2  5 x  6)
x 2  5x  6
( x  6)( x  1)( x  2)( x  3) ( x  1)( x  2)

( x  3)( x  2)( x  6)( x  1) ( x  2)( x  1)
Adding and Subtracting Rational Expressions
- Adding and Subtracting Rational Expressions with the same denominator
To add/subtract two (or more) rational expressions with the same denominator
- add/subtract the numerators and keep the denominator, a  c  (a)  (c)
b b
b
- reduce to the lowest terms
Example:
3x  5 2 x  4 (3x  5)  (2 x  4) 3x  5  2 x  4 x  9
a)




2x  1 2x  1
2x  1
2x  1
2x  1
6
x
6
x
6
x
6 x
b)






x  1 1  x x  1  ( x  1) x  1 x  1 x  1
- Adding and Subtracting Rational Expressions with different denominators
(i) Find the LCD of all rational expression:
a) Factor each denominator completely; write multiple factors as exponents
b) Identify different factors in all denominators
c) Form the LCD: LCD is the product of all different factors raised to the largest exponent
that appears in any denominator
(ii) Write each rational expression as an equivalent expression with the denominator =LCD
(iii) Add/Subtract the numerators and keep the denominator
(iv) Reduce to lowest terms
Example: a)
(i)
2x
3

9 y 3 4 xy
LCD:
9y3 = 32y3
4xy= 22x y
Different factors: 3,y,2,x.
(ii)
(iii)
LCD = 32y322x = 9∙4 xy3
2x
3
2x  4x
39y2
8x 2
27 y 2





9 y 3 4 xy 9 y 3  4 x 4 xy  9 y 2 36 xy 3 36 xy 3
2x
3
2x  4x
39y2
8x 2
27 y 2 8 x 2  27 y 2






9 y 3 4 xy 9 y 3  4 x 4 xy  9 y 2 36 xy 3 36 xy 3
36 xy 3
b)
x 1
3x  1
x 1
3x  1
( x  1)( x  4)
(3x  1) x
 2





find x ( x  4)( x  4)
( x  4)( x  4) x
x  4 x x  16 factor x( x  4) ( x  4)( x  4) LCD
2
x 2  5x  4
3x 2  x
( x 2  5 x  4)  (3x 2  x)



x( x  4)( x  4) x( x  4)( x  4)
x( x  4)( x  4)
x 2  5 x  4  3x 2  x  2 x 2  6 x  4  2( x 2  3x  2)  2 x 2  6 x  4



x( x  4)( x  4)
x( x  4)( x  4)
x( x  4)( x  4)
x( x  4)( x  4)
Complex Fraction: a rational expression that contains rational expression(s) in the numerator and/or the denominator.
You MUST always simplify a complex fraction
Method 1: (i) Perform operations in the numerator and the denominator (add/subtract fractions)
( ii) Divide the expression in the numerator by the expression in the denominator and simplify
Example
x
5
3 
4 1

x 2x
53 x
15  x 15  x

2
1  3 3  3  3  15  x  7  15  x  2 x  2 x(15  x)  2 x  30 x
42 1
8 1
7
3
2x
3
7
21
21

x  2 2x
2x
2x
Method 2. (i) Find the LCD of all fractions in the numerator and the denominator
(ii)Multiply all terms in the numerator and the denominator by the LCD
(iii)Simplify
Example: Simplify
x
3
4 1

x 2x
5
(i)
Find the LCD of x , 4 , 1 . The LCD = 6x
3 x 2x
(ii)
Multiply every term by the LCD (6x)
x
x
x 6x
5
5  6x   6x
30 x  
2
2
3 
3
3 1  30 x  2 x  30 x  2 x

4 1
4
1
4 6x 1 6x
24  3
21

 6x 
 6x



x 2x x
2x
x 1 2x 1
R.8 n-th Radicals; Rational Exponents
A square root of a nonnegative number a is a number b such that b2 = a.
Example : square roots of 25 are 5 and -5 since 52 = 25 and (-5)2 = 25
Square root of 0 is 0, since 02 = 0
Square root of – 4 does not exist (in the real number system), since there is no real
number that squared gives (-4)
The principal square root, or radical
number a is called a radicand.
Example
25  5 ,
0  0,
, of a nonnegative number a is a nonnegative number b such that b2 = a. The
 4 not defined
Remark : a) the principal square root of a is often called the square root of a or radical of a
b) the square root of a is NEVER negative
Properties of radicals
 a
2
 a,
 3
2
a0
(5) 2 | 5 | 5
a 2 | a |
a  b  a  b,
a

b
am 
a
b
3
95  9  5  3 5
a, b  0
4

25
b0
,
 a ,
m
4
25
16 3 
a0

 16 
3
2
5
 4 3  64
To simplify a radical means to remove all factors that are perfect squares
 
Example : 12 x 5  3  4  x 4  x  4  x 2
Note that
x8 
x 
4 2
2
x 
2 2
 3x  4
 x 4 and, in general,
x 5  x 4  x  x 2 x and, in general,
 3x  2 x 2 3x
x even  x even / 2
x odd  x (odd1) / 2 x
If two expressions contain the same radical (same index and same radicand), then that radical can be factored out and
the two expressions combined.
Example: 3 12  4 27  3 4  3  4 9  3  3  2 3  4  3 3  6 3  12 3  3(6  12)  6 3
To rationalize the denominator (or numerator) is to eliminate the radical from the denominator (or numerator) through
some algebraic operations
Denominator is
Multiply the numerator
and the denominator by
a x
The denominator becomes
a x  x  ax
x
a  b x a  b x   a  b x 
ab x
ab x
ab x
ab x
a u b w
a u b w
a u b w
a u b w
2
a

2
 a 2  b2 x
   
2

2
u  b w a u  b w  a u  b w  a 2u  b 2 w
Example: Rationalize the denominator
a)  3   3  5   15   15
25
10
2 5 2 5 5
b) 2  3  2  3  1  5  2  2 5  3  3 5  2  2 5  3  15
2
4
1 5
1 5  1 5
12  5






 
Higher order radicals and rational exponents
If n > 2 then
a , is such a number b that bn = a.
-if n is an even number, then the n-th radical of a nonnegative number a, denoted n a , is such a nonnegative number b,
-if n is an odd number, then the n-th radical of a, denoted
that bn = a.
n
Example
32  2 since 25 = 32; 3  27  3 since (-3)3 = -27
4
256  4 since 44 = 256; 4  16 not defined
In the notation n a , a is called the radicand and n is called the index of a radical.
5
n-th radical has similar properties as the square root
(i)
(ii)
(iii)
 a
n
n
a
a n  a if n is odd
n
n
and
n
a n | a | if n is even
a  b  n a  n b provided all radicals exist
a na

(iv)
provided all radicals exist and b  0
b nb
n
(v)
am 
n
 a
m
n
provided all radicals exist
To simplify n-th radical is to remove from the radical any perfect roots (using property (ii)), which means that none of
the exponents inside the radical can be greater or equal to the index n.
Example
4
x5 y 7
x 4 xy 4 y 3


4
4
z8
z2
 
4
x 4 y 4 4 xy 3
4
z 
2 4

| x || y | 4 xy 3
z2
Rational exponents:
If n is a natural number then
1
n
a  n a , provided that n a
If m is an integer and n is a natural number then a
Example
m
n
exists
 n am 
 a
n
m
1
42  2 4  4  2
1
(8) 3  3  8  2
3
2 2  2 23  8  2 2
2
1
1
1
8 3  3 8 2  3 2  3

8
64 4
1
(25) 2  2  25   25 not defined (not a real number)
Rational exponents have the same properties as the integer exponents:
If a and b are real numbers and r and t are rational then
(i) ar  at  ar t
ar
(ii) t  a r t
a
t
(iii) a r   a r t
(iv) a  br  at  br
a
b
r
(v)   
ar
br
b 0
provided all exponents are defined
Example: Simplify
4 x
1

y1/ 3
( xy ) 3 / 2
3/ 2

4 3 / 2 x 3 / 2 y 1 / 2
8
8
 3 / 23 / 2 3 / 21 / 2  3
3/ 2 3/ 2
x y
x
y
x y