8

1. UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra

2. Certain Basic Theorems
UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra
Theorem 1. Cancellation Law for
Addition. If a+c=b+c then a=b
Theorem 2. Cancellation Law for
Multiplication. If ac=bc, c≠0 then
a=b.

3. Certain Basic Theorems
UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra
Theorem 3. If a=b then -a=-b (If two
numbers are equal then their negatives are also equal)
Theorem 4. For any real number
a, ax0=0. (The product of any real number and
zero is equal to zero)

4. Certain Basic Theorems
UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra
Theorem 5. ab=0 if and only if
a=0 or b=0.(Whenever the product of two
numbers is zero, one or both of the numbers is zero)
Theorem 6. –(-a) = a. (The negative of
negative of a is equal to a.)

5. Certain Basic Theorems
UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra
Theorem 7. For any non-zero real
number a, 1/(1/a) =a.(The reciprocal of
the reciprocal of a non-zero number is equal to the
non-zero number a)

6. Theorems involving Subtraction and
Division
UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra
Definition. Subtraction.
Subtracting b from a denoted by
a-b, is equal to adding the
negative of b to a.That is, a-b=a+(-b)

7. Theorems involving Subtraction and
Division
UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra
Theorem 8. a(b-c)=ab-ac (Distributive
axiom).Theorem 9.
-(a+b)=-a-b
-(a-b)=-a+b

8. Theorems involving Subtraction and
Division
UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra
Definition. Division. The quotient
of a by b, written as a/b, is
equivalent to multiplying a by the
reciprocal of b, b≠0.
That is, a/b=a (1/b), b≠0.

9. Theorems involving Subtraction and
Division
UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra
Theorem 10. For any non-zero
real number a, a/1 =a.
For any real number b, b/1 =b.
Theorem 11.
, 0
a b a b
c
c c c

  

10. Theorems involving Subtraction and
Division
UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra
Theorem 12.
, 0
ac a
c
bc b
 
Theorem 13.
, , 0
a c ac
b d
b d bd
  

11. Theorems involving Subtraction and
Division
UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra
Theorem 14.
Theorem 15.
, 0, 0
a c ad bc
b d
b d bd

   
, 0, 0
a c
iffad bc b d
b d
   

12. Theorems involving Subtraction and
Division
UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra
Theorem 16.
Theorem 17. for any real numbers
a and b, b≠0,
, , , 0
a c a d
b c d
b d b c
   
a a a a
and
b b b b
  
   
 

13. UNIVERSITY OF THE PHILIPPINES MINDANAO
College Algebra
1. College Algebra and
Trigonometry by Leithold
2. Mathematics Modules. Summer
Bridge Program, Summer 200 by
UP-LRC
3. Workbook for PSI for College
Algebra by UP-LRC