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Chapter 4
Products and Factors
of Polynomials
Section 4-1
Polynomials
Constant  A number
 -2, 3/5, 0
Monomial A constant, a variable,
or a product of a
constant and one or
more variables
 -7
5u
(1/3)m2
-s2t3
x
Coefficient The constant (or
numerical) factor in a
monomial
 3m2
coefficient = 3
u
coefficient = 1
 - s2t3
coefficient = -1
Degree of a Variable The number of times the
variable occurs as a factor
in the monomial
 For Example – 6xy3
What is the degree of x? y?
Degree of a monomial The sum of the degrees of
the variables in the
monomial. A nonzero
constant has degree 0.
 The constant 0 has no
degree.
Examples 6xy3
 -s2t3
u
 -7
degree = 4
degree = 5
degree = 1
degree = 0
Similar Monomials Monomials that are
identical or that differ only
in their coefficients
 Also called like terms
 Are - s2t3 and 2s2t3 similar?
Polynomial A monomial or a sum of
monomials.
 The monomials in a
polynomial are called the
terms of the polynomial.
Examples x2 + (-4)x + 5
 x2 – 4x + 5
 What are the terms?
2
 x , -4x, and 5
Simplified Polynomial A polynomial in which no
two terms are similar.
 The terms are usually
arranged in order of
decreasing degree of one
of the variables
Are they Simplified?
 2x3 – 5 + 4x + x3
 3x3 + 4x – 5
 4x2 – x + 3x4 – 5 + x2
Degree of a Polynomial The greatest of the
degrees of its terms after
it has been simplified
 What is the degree?
x4 + 3x
2x3 + 3x – 7
2
x – 5x + 1 7x + 1
x4 – 2x2y3 + 6y -11
Adding Polynomials
 To add two or more
polynomials, write their
sum and then simplify by
combining like terms
 Add the following-
(x2 + 4x – 3) +
(x3 – 2x2 + 6x – 7)
Subtracting Polynomials
 To subtract one
polynomial from another,
add the opposite of each
term of the polynomial
you’re subtracting
3
 (x
–
2
5x
+ 2x – 5) –
(2x2 – 3x + 5)
Section 4-2
Using Laws of
Exponents
Laws of Exponents
 Let a and b be real
numbers and m and n
be positive integers in
all the following laws
Law 1
 am · an = am+n
 x2 · x 4 = x 6
3
y
5
y
·
=?
 m · m4 = ?
Law 2
 (ab)m = ambm
 (xy)3 = x3y3
2
 (3st)
=?
 (xy)5 = ?
Law 3
 (am)n = amn
 (x3)2 = x6
2
3
4
 (x y )
=?
 (2mn2)3 = ?
Using Distributive Law
 Distribute the variable
using exponent laws
 3t2(t3 – 2t2 + t – 4) = ?
 – 2x2(x3 – 3x + 4) = ?
Section 4-3
Multiplying
Polynomials
Binomial
 A polynomial that has
two terms
2x + 3
3xy – 14
4x – 3y
613 + 39z
Trinomial
 A polynomial that has
three terms
2x2 – 3x + 1
14 + 32z – 3x
mn – m2 + n2
Multiplying binomials
 When multiplying two
binomials both terms of
each binomial must be
multiplied by the other
two terms
 Using the F.O.I.L method
helps you remember the
steps when multiplying
F.O.I.L. Method
 F – multiply First terms
 O – multiply Outer terms
 I – multiply Inner terms
 L – multiply Last terms
 Add all terms to get
product
Binomial
 A polynomial that has
two terms
2x + 3
3xy – 14
4x – 3y
613 + 39z
Trinomial
 A polynomial that has
three terms
2x2 – 3x + 1
14 + 32z – 3x
mn – m2 + n2
Multiplying binomials
 When multiplying two
binomials both terms of
each binomial must be
multiplied by the other
two terms
 Using the F.O.I.L method
helps you remember the
steps when multiplying
F.O.I.L. Method
 F – multiply First terms
 O – multiply Outer terms
 I – multiply Inner terms
 L – multiply Last terms
 Add all terms to get
product
Example - (2a – b)(3a + 5b)
 F – 2a · 3a
 O – 2a · 5b
 I – (-b) ▪ 3a
 L - (-b) ▪ 5b
2
2
 6a + 10ab – 3ab – 5b
 6a2 + 7ab – 5b2
Example – (x + 6)(x +4)
F – x ▪ x
O – x ▪ 4
I – 6 ▪ x
L – 6 ▪ 4
 x2 + 4x + 6x + 24
 x2 + 10x + 24
Special Products
 (a + b)2 = a2 + 2ab + b2
 (a - b)2 = a2 - 2ab + b2
 (a + b)(a – b) =
2
a
-
2
b
Section 4-4
Using Prime
Factorization
Factor
 A number over a set of
numbers, you write it as a
product of numbers
chosen from that set
 The set is called a factor
set
Example
 The number 15 can be
factored in the
following ways
(1)(15)
(5)(3)
(-1)(-15)
(-3)(-5)
Prime Number
 An integer greater than 1
whose only positive
integral factors are itself
and 1
Prime Factorization
 If the factor set is
restricted to the set of
primes
 To find it you write the
integer as a product of
primes
Example
 350 = 2 x 175
= 2 x 5 x 35
=2x5x5x7
 So the prime factorization
of 350 is 2 x 52 x 7
Greatest Common Factor
 The greatest integer that
is a factor of each
number.
 To find the GCF, take the
least power of each
common prime factor.
Example
 What is the GCF of
100, 120, and 90?
 10
Least Common Multiple
 The least positive
integer having each as
a factor
 To find the LCM, take
the greatest power of
each common prime
factor.
Example
 What is the LCM of
100, 120, and 90?
1800
Summary
 GCF – take the least
power of each common
prime factor.
 LCM – take the greatest
power of each prime
factor
Section 4-5
Factoring
Polynomials
Factor
 To factor a polynomial
you express it as a
product of other
polynomials
 We will factor using
polynomials with
integral coefficients
Greatest Monomial Factor
 The GCF of the terms
 What is the GCF of
4
3
2
2x – 4x + 8x ?
2
 2x
Now factor:
 2x4 – 4x3 + 8x2
 Factor out 2x2
2
2
 2x (x – 2x + 4)
Perfect Square Trinomials
 The polynomials in the
form of a2 + 2ab + b2
2
2
and a – 2ab + b are the
result of squaring a + b
and a – b respectively
Difference of Squares
 The polynomial a2 – b2 is
the product of a + b and
a-b
Factor Each Polynomial
 z2 + 6z + 9
 4s2 – 4 st + t2
2
2
 25x – 16a
Factored Form:
 (z + 3)2
 (2s – t)2
 (5x + 4a)(5x – 4a)
Sum and Difference of Cubes
 a3 + b 3 =
(a + b)(a2 - ab + b2)
3
a
3
b
–
=
(a – b)(a2 + ab + b2)
Factor Each Polynomial
 y3 - 1
 8u3 + v3
Factor by Grouping
 Factor each polynomial by
grouping terms that have
a common factor
 Then factor out the
common factor and write
the polynomial as a
product of two factors
Factor each Polynomial
 3xy - 4 - 6x + 2y
 xy + 3y + 2x + 6
Section 4-6
Factoring Quadratic
Polynomials
Quadratic Polynomials
 Polynomials of the
2
form ax + bx + c
 Also called seconddegree polynomials
Terms
- quadratic term
bx - linear term
c - constant term
2
ax
Quadratic Trinomial
 A quadratic
polynomial for which
a, b, and c are all
nonzero integers
Factoring Quadratic
Trinomials
 ax2 + bx + c can be
rewritten in another
format and then
factored by grouping.
Factors
 Multiply ac
 Find factors ac that
add to b
 Rewrite the trinomial
Factor the Polynomial
 x2 + 2x – 15
 a = 1, c= – 15
 Factors of ac that add
to b are 5 and – 3
 x2 + 5x – 3x – 15
Factor the Polynomial
 x2 + 5x – 3x – 15
 Factor by grouping
2
 (x + 5x) + (– 3x – 15)
 x(x + 5) –3(x + 5)
 (x+5)(x – 3)
Factor the Polynomials
2
15t
- 16t + 4
3 - 2z 2
x
2
z
+ 4x - 3
Irreducible
 If a polynomial has more
than one term and cannot
be expressed as a
product of polynomials of
lower degree taken from
a given factor set, it is
irreducible
2
 x + 4x - 3 is irreducible
Factored Completely
 A polynomial is factored
completely when it is
written as a product of
factors and each factor is
either a monomial, a
prime polynomial, or a
power of a prime
polynomial
Greatest Common Factor
 The GCF of two or more
polynomials is the
common factor having
the greatest degree and
the greatest constant
factor
Least Common Multiple
 The LCM of two or more
polynomials is the
common multiple
having the least degree
and least positive
constant factor
Section 4-7
Solving Polynomial
Equations
Polynomial Equation
 An equation that is
equivalent to one with a
polynomial as one side
and 0 as the other
2
x
= 5x + 24
Root
 The value of a variable
that satisfies the
equation
 Also called the solution
Solving a polynomial
Equation
 You can factor the
polynomial to solve the
equation
Steps to Solving a
polynomial Equation
 Write the equation with 0
as one side
 Factor the other side of
the equation
 Solve the equation
obtained by setting each
factor equal to 0
Example 1
 Solve (x – 5)(x + 2) = 0
Step 1: already = 0
Step 2: already factored
Step 3: set each factor = 0
x-5=0
x+2=0
x=5
x = -2
Example 2
 Solve x2 = x + 30
1: x2 - x – 30 = 0
2: (x – 6)(x + 5) = 0
3: x – 6 = 0
x+5=0
x=6
x = -5
The solution set is {6, -5}
Zeros
 A number r is a zero of a
function f if f(r) = 0
 You can find zeros using
the same method that is
used to solve polynomial
equations
Example
 Find the zeros of
f(x) = (x – 4)3 – 4(3x – 16)
1: simplify
2: factor
3: set each factor = 0
Double Zero
 A number that occurs
as a zero of a function
twice
Double Root
 A number that occurs
twice as a root of a
polynomial equation
Solve
 x2 + 25 = 10x
 12 + 4m = m2
Section 4-8
Problem Solving Using
Polynomial Equations
Example 1
 A graphic artist is designing a
poster that consists of a
rectangular print with a
uniform border. The print is
to be twice as tall as it is
wide, and the border is to be
3 in. wide. If the area of the
poster is to be 680 in2, find
the dimensions of the print.
Solution
1.Draw a diagram
2.Let w = width and
2w = height
1.The dimensions are 6 in.
greater than the print,
so they are w + 6 and
2w + 6
Solution
4. The area is
represented by
(w + 6)(2w + 6) = 680
5. Solve the equation.
Example 2
 The sum of two
numbers is 9. The sum
of their squares is 101.
Find the numbers.
Solution
1.Let x = one number
2.Then 9 – x = the other
number
3.x2 + (9 – x)2 = 101
Section 4-9
Solving Polynomial
Inequalities
Polynomial Inequality
 An inequality that is
equivalent to an
inequality with a
polynomial as one side
and 0 as the other side.
 x2 > x + 6
Solve by factoring
 The product is positive if
both factors are positive,
or both factors are
negative
 The product is negative if
the factors have opposite
signs
Example 1
 Solve and graph
x2 – 1 > x + 5
 x2 – x – 6 > 0
 Both factors must be
positive or negative
Example 2
 Solve and graph
 3t < 4 – t2
 t2 + 3t – 4 < 0
 The factors must have
opposite signs