Intermediate Algebra 098A
Review of
Exponents &
Factoring
1.1 – Integer Exponents
• For any real number b and any natural
number n, the nth power of b is found by
multiplying b as a factor n times.
b  bbb
n
b
Exponential Expression – an
expression that involves
exponents
• Base – the number being multiplied
• Exponent – the number of factors of the
base.
Product Rule
a a a
n n
mn
Quotient Rule
m
a
mn

a
n
a
Integer Exponent
a
n
1
 n
a
Zero as an exponent
a  1 a  0  R
0
Calculator Key
• Exponent Key
^
Sample problem
3
0
8x y
2 5
24 x y
5
y
 5
3x
more exponents
• Power to a Power
a


n
m
a
mn
Product to a Power
 ab 
r
a b
r r
Polynomials - Review
•Addition
•and
• Subtraction
Objective:
• Determine the
coefficient and degree
of a monomial
Def: Monomial
• An expression that is a constant
or a product of a constant and
variables that are raised to
whole –number powers.
• Ex: 4x 1.6 2xyz
Definitions:
• Coefficient: The numerical
factor in a monomial
• Degree of a Monomial: The
sum of the exponents of all
variables in the monomial.
Examples – identify the degree
8x
4
0.5x y
4
5
4
5
Def: Polynomial:
• A monomial or an
expression that can be
written as a sum or
monomials.
Def: Polynomial in one variable:
• A polynomial in which
every variable term has
the same variable.
Definitions:
• Binomial: A polynomial
containing two terms.
• Trinomial: A
polynomial containing
three terms.
Degree of a Polynomial
• The greatest degree of
any of the terms in the
polynomial.
Examples:
5 x  x  10 x  9 x  1
6
3
2
3x  4 x  5
2
x  16
2
x  3x y  2 xy  y  6
5
3
4
3
2
Objective
•Add
•and
•Subtract
• Polynomials
To add or subtract Polynomials
• Combine Like Terms
• May be done with columns or
horizontally
• When subtracting- change the
sign and add
Evaluate Polynomial Functions
• Use functional notation to
give a polynomial a name
such as p or q and use
functional notation such as
p(x)
• Can use Calculator
Calculator Methods
•
•
•
•
•
•
1.
2.
3.
4.
5.
6.
Plug In
Use [Table]
Use program EVALUATE
Use [STO->]
Use [VARS] [Y=]
Use graph- [CAL][Value]
Objective:
• Apply evaluation of
polynomials to real-life
applications.
Intermediate Algebra 5.4
•Multiplication
•and
•Special Products
Objective
• Multiply
•a
• polynomial
• by a
• monomial
Procedure: Multiply a
polynomial by a monomial
• Use the distributive property to
multiply each term in the
polynomial by the monomial.
• Helpful to multiply the
coefficients first, then the
variables in alphabetical order.
Law of Exponents
r s
b b b
r
s
Objectives:
• Multiply Polynomials
• Multiply Binomials.
• Multiply Special
Products.
Procedure: Multiplying
Polynomials
• 1. Multiply every term in the
first polynomial by every term
in the second polynomial.
• 2. Combine like terms.
• 3. Can be done horizontally or
vertically.
Multiplying Binomials
• FOIL
• First
• Outer
• Inner
• Last
Product of the sum and difference
of the same two terms
Also called multiplying
conjugates
 a  b a  b  a
2
b
2
(a  b)  a 2  ab  b 2 
Squaring a Binomial
 a  b   a  2ab  b
2
2
2
 a  b   a  2ab  b
2
2
2
Objective:
• Simplify Expressions
• Use techniques as part of a
larger simplification
problem.
Albert EinsteinPhysicist
• “In the middle of
difficulty lies
opportunity.”
Intermediate Algebra –098A
•Common Factors
•and
• Grouping
Def: Factored Form
• A number or
expression written as a
product of factors.
Greatest Common Factor (GCF)
• Of two numbers a and b is the
largest integer that is a factor of
both a and b.
Calculator and gcd
• [MATH][NUM]gcd(
• Can do two numbers – input
with commas and ).
• Example: gcd(36,48)=12
Greatest Common Factor (GCF)
of a set of terms
•Always do this
FIRST!
Procedure: Determine greatest common
factor GCF of 2 or more monomials
• 1. Determine GCF of numerical
coefficients.
• 2. Determine the smallest
exponent of each exponential
factor whose base is common to
the monomials. Write base with
that exponent.
• 3. Product of 1 and 2 is GCF
Factoring Common Factor
Find the GCF of the terms
• 2. Factor each term with the
GCF as one factor.
• 3. Apply distributive property
to factor the polynomial
• 1.
Example of Common Factor
16 x y  40 x 
3
2
8 x (2 xy  5)
2
Factoring when first terms is
negative
• Prefer the first term inside parentheses to be
positive. Factor out the negative of the
GCF.
20 xy  36 y 
3
4 y (5 xy  9)
2
Factoring when GCF is a
polynomial
a(c  5)  b(c  5) 
(c  5)(a  b)
Factoring by Grouping – 4 terms
• 1. Check for a common factor
• 2. Group the terms so each group has a
common factor.
• 3. Factor out the GCF in each group.
• 4. Factor out the common binomial factor –
if none , rearrange polynomial
• 5. Check
Example – factor by grouping
32 xy  48 xy  20 y  30 y 
2
2
2 y 16 xy  24 x  10 y  15 
2 y  2 y  38 x  5
Ralph Waldo Emerson – U.S.
essayist, poet, philosopher
•“We live in
succession , in
division, in parts, in
particles.”
Intermediate Algebra 098A
•Special Factoring
Objectives:Factor
• a difference of squares
• a perfect square trinomial
• a sum of cubes
• a difference of cubes
Factor the Difference of two
squares
a  b   a  b  a  b 
2
2
Special Note
• The sum of two squares is
prime and cannot be factored.
a b
2
2
is prime
Factoring Perfect Square
Trinomials
a  2ab  b   a  b 
2
2
a  2ab  b   a  b 
2
2
2
2
Factor: Sum and Difference of
cubes
a  b  (a  b)  a  ab  b
2
a  b  (a  b)  a  ab  b
2
3
3
3
3
2
2


Note
• The following is not factorable
a  ab  b
2
2
Factoring sum of Cubes informal
• (first + second)
• (first squared minus first times
second plus second squared)
Intermediate Algebra 098A
• Factoring Trinomials
• of
• General Quadratic
ax  bx  c
2
 50 y  15 y 
Objectives:
• Factor trinomials of the form
x  bx  c
2
ax  bx  c
2
Factoring
x  bx  c
2
• 1. Find two numbers with a product equal
to c and a sum equal to b.
• The factored trinomial will have the form(x
+ ___ ) (x + ___ )
• Where the second terms are the numbers
found in step 1.
• Factors could be combinations of positive
or negative
Factoring
Trial and Error
ax  bx  c
2
• 1. Look for a common factor
• 2. Determine a pair of coefficients of first
terms whose product is a
• 3. Determine a pair of last terms whose
product is c
• 4. Verify that the sum of factors yields b
• 5. Check with FOIL Redo
Factoring ac method
ax  bx  c
2
• 1. Determine common factor if any
• 2. Find two factors of ac whose sum is b
• 3. Write a 4-term polynomial in which by
is written as the sum of two like terms
whose coefficients are two factors
determined.
• 4. Factor by grouping.
Example of ac method
6x 11x  4 
2
6x  3x  8x  4 
2
3x(2x 1)  4(2x 1) 
(2x 1)(3x  4)
Example of ac method
5 y (8 y  10 y  3) 
2
2
5 y 8 y  2 y  12 y  3 
2
2
5 y 2 y  4 y  1  3 4 y  1 
2
5 y  4 y  1 2 y  3
2
Factoring - overview
•
•
•
•
•
•
•
1. Common Factor
2. 4 terms – factor by grouping
3. 3 terms – possible perfect square
4. 2 terms –difference of squares
Sum of cubes
Difference of cubes
Check each term to see if completely
factored
Isiah Thomas:
• “I’ve always believed no
matter how many shots I
miss, I’m going to make
the next one.”
Intermediate Algebra 098A
•Solving Equations
•by
•Factoring
Zero-Factor Theorem
•If a and b are real
numbers
•and ab =0
•Then a = 0 or b = 0
Example of zero factor property
 x  5 x  2   0
x  5  0 or x  2  0
x  5 or x  2
5, 2
or
2, 5
Solving a polynomial equation by
factoring.
1. Factor the polynomial
completely.
2. Set each factor equal to 0
3. Solve each of resulting equations
4. Check solutions in original
equation.
5. Write the equation in standard
form.
Example – solve by factoring
3x  11x  4
2
3x 11x  4  0
2
 3x  1 x  4   0
3x  1  0 or
x4 0
1
x
or x  4
3
Example: solve by factoring
x  4 x  12 x
3
2
x  4 x  12 x  0
3
2
x  x  4 x  12   0
2
x  x  6  x  2   0
0, 6, 2
Example: solve by factoring
• A right triangle has a
hypotenuse 9 ft longer than the
base and another side 1 foot
longer than the base. How long
are the sides?
• Hint: Draw a picture
• Use the Pythagorean theorem
Solution
x   x  1   x  9
2
2
2
x  20 or x  4
• Answer: 20 ft, 21 ft, and 29 ft
Example – solve by factoring
3x  2 x  7   12
• Answer: {-1/2,4}
Example: solve by factoring
1 2
1
1 2
x  3  x   x  2 

2
12
3
• Answer: {-5/2,2}
Example: solve by factoring
9 y  y  1  4 y  6 y  1  3 y
2
• Answer: {0,4/3}
Example: solve by factoring
t  3t  13  7t   3t  1
3
2
• Answer: {-3,-2,2}
Sugar Ray Robinson
• “I’ve always believed that
you can think positive just
as well as you can think
negative.”
Maya Angelou - poet
• “Since time is the one
immaterial object which we
cannot influence – neither
speed up nor slow down, add
to nor diminish – it is an
imponderably valuable gift.”