Download polynomials

POLYNOMIALS
I. Exponents
A. Definitions

Exponents are shorthand notation for repeated multiplication
a n  a 
a
 a
 
a.
n factors

Zero exponent b 0  1

Negative exponent b  n 
1
bn
To see why the zero exponent and negative exponents are defined this way
consider the pattern in the example below. In the left column the exponents
decrease by 1 and in the right column the numbers decrease by a factor of 10.
103  1000
10 2  100
101  10
10 0  1
1
1
 1
10 10
1
1

 2
100 10
1
1

 3
1000 10
10 1 
10  2
10 3
B. Rules of Exponents
Product rule (same base)
Quotient rule (same base)
Product rule (same exponent)
a ma n  a m n
am
m n
n  a
a
( ab) m  a mb m
m
Quotient rule (same exponent)
am
 a
   m
 b
b
Power rule
a 
m n
 a mn
II. Polynomials
A polynomial has terms consisting of constants and variables raised to
positive integer powers. Polynomials are classified according to the
number of terms they have. Monomials have one term, binomials have two
terms and trinomials have three terms. The degree of a polynomial in one
variable is the greatest exponent. The degree of a polynomial of several
variables is the greatest sum of the exponents that occurs in any of the
terms.
III. Addition/Subtraction
Like terms have the same variables with the same exponents.
Polynomials are added/subtracted by combining like terms.
IV. Multiplication
A. Multiply a polynomial by a monomial
Multiply each term of the polynomial by the monomial. This is the same as
saying use the distributive rule.
B. Multiply a polynomial by a polynomial
FOIL Method.
When multiplying binomials use first, outer, inner and last and combine
like terms.
General Method.
Multiply each term of the second polynomial by each term of the first
polynomial and combine like terms.
V. Factoring
Choose a factoring technique based on the number of terms.
A. Greatest common factor
Always factor out the GCF before moving on to other techniques. Factor
out the largest number that divides into each term evenly. Factor out the
variables that are common to all the terms using the lowest exponent.
B. Factoring formulas (two terms)
Difference of squares
Difference of cubes
Sum of cubes
x 2  a 2  ( x  a )( x  a )
x 3  a 3  ( x  a )( x 2  ax  a 2 )
x 3  a 3  ( x  a )( x 2  ax  a 2 )
C. Factoring trinomials (three terms)
When the polynomial has three terms use the trial and error method.
Leading coefficient is a 1.
Since x 2  (a  b) x  ab  ( x  a)( x  b) to factor x 2  bx  c we must find
numbers whose sum is b and whose product is c. In other words find two
numbers that multiply to the last term and add to give the coefficient of
the middle term.
Leading coefficient is not a 1.



Firsts must multiply to give the first term.
Lasts must multiply to give the third term.
The inner and outer must sum to give the second term.
Comments:
(1) The AC test states that ax 2  bx  c is factorable only if there are two
numbers whose product is ac and whose sum is b. For example
3 x 2  2 x  5 is not factorable since there are no numbers, which multiply
to 15 and add to 2.
(2) The slide technique as shown in the example below has less trial and
error for the case in which the leading coefficient is not 1.
6 x 2  x  12
x 2  x  72
9 
8

 x   x  
6 
6

3 
4

 x   x  
2 
3

(2 x  3)(3 x  4)
Multiply the last term by the
leading coefficient.
Factor and then divide both
numbers
by
the leading
coefficient.
Reduce the fractions.
Multiple the denominators up
to the variables.
D. Factoring by grouping (four or more terms)
Rearrange the terms (if necessary) so that the first two terms have a
common factor and the last two terms have a common factor. Factor out the
GCF from the first two terms and the second two terms. Factor a binomial
factor from the remaining terms.
VI. Solving Equations
Use the zero factor property. If ab  0 then a  0 or b  0 .
VII. Division
A. Division by a monomial
Divide each term by the monomial
B. Division by a polynomial
Use the long division algorithm.
C. Synthetic division
Used to divide a polynomial by a binomial of the form x  c .
Ex. Divide 2 x 4  3x 2  5x  7 by x  3 .
3 2
0
6
2 6
3
5 7
18  45 120
15  40 113
2 x 3  6 x 2  15 x  40 
113
x3
College Algebra Home Page