Chapter 11
Polynomials
11-1
Add & Subtract
Polynomials
Monomial
A constant, a variable, or a
product of a constant and
one or more variables
-7
5u
2
(1/3)m
2
3
-s t
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
Polynomial
Expressions with several
terms that follow
patterns.
4x3 + 3x2 + 15x + 2
3b2 – 2b + 4
Coefficient
 The
constant (or
numerical) factor in a
monomial
 3m2
coefficient = 3
u
coefficient = 1
 -s2t3
coefficient = -1
Like Terms
 Terms
that are identical
or that differ only in their
coefficients
 Are 2x and 2y similar?
 Are -3x2 and 2x2 similar?
Examples
2
x
+ (-4)x + 5
2
 x – 4x + 5
 What are the terms?
 x2, -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?
2
 2x
2
x
– 5 + 4x +
 3x + 4x – 5
 4x2 – x + 3x2 – 5 + x2
11-2
Multiply by a
Monomial
Examples
 (5a)(-3b)
2
2
 3v (v
+ v + 1)
 12(a2 + 3ab2 – 3b3 – 10)
11-3
Divide and Find
Factors
GREATEST COMMON
FACTOR
The greatest integer
that is a factor of all
the given integers.
2,3,5,7,11,13,17,19,23,29
Prime number - is an
integer greater than
1 that has no positive
integral factor other
than itself and 1.
GREATEST COMMON
FACTOR
Find the GCF of 25 and 100
25 = 5 x 5
100 = 2 x 2 x 5 x 5
GCF = 5 x 5 = 25
GREATEST COMMON
FACTOR
Find the GCF of 12 and 36
12 =
36 =
GCF =
GREATEST COMMON
FACTOR
Find the GCF of 14,49 and 56
14 =
49 =
56 =
GCF =
Factoring Polynomials
vw + wx
= w(v + x)
Factoring Polynomials
2
21x
=
–
2
35y
Factoring Polynomials
13e – 39ef
=
Dividing Polynomials by
Monomials
5m + 35
5
= 5(m+ 7)÷5
= m+7
Dividing Polynomials by
Monomials
7x + 14
7
= 7x + 14
7
7
= x+2
Dividing Polynomials by
Monomials
6a + 8b
2
= 2(a +4b) ÷ 2
= a + 2b
Dividing Polynomials by
Monomials
2
6x
2x +
2x
11-4
Multiply Two
Binomials
Multiplying Binomials
When multiplying two
binomials both terms
of each binomial must
be multiplied by the
other two terms
Multiplying binomials
 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
Example:
(x + 6)(x +4)
F
–x▪x
O – x ▪ 4
I – 6 ▪ x
L – 6 ▪ 4
11-5
Find Binomial
Factors in a
Polynomial
Procedure
Group the terms in the
polynomial as pairs that
share a common monomial
factor
• Extract the monomial
factor from each pair
•
Procedure
•
•
If the binomials that remain for
each pair are identical, write this
as a binomial factor of the whole
expression
The monomials you extracted
create a second polynomial. This
is the paired factor for the original
expression
Example
4x3 + 4x2y2 + xy + y3
Group (4x3 + 4x2y2) and factor
Group (xy + y3) and factor
4x2(x +y2) + y(x + y2)
Answer: (x +y2) (4x2 + y)
Example
2x3 - 2x2y - 3xy2 + 3y3+ xz2 – yz2
Group (2x3 - 2x2y2 ) and factor
Group (- 3xy2 + 3y3) and factor
Group (xz2 – yz2) and factor
Answer:
11-6
Special Factoring
Patterns
11-6 Difference of
Squares
(a + b)(a – b)=
2
a
(x + 5) (x – 5) =
-
2
x
2
b
- 25
11-6 Squares of
Binomials
(a +
2
b)
=
2
a
+ 2ab +
2
b
(a - b)2 = a2 - 2ab + b2
•
Also known as Perfect
square trinomials
Examples
(x +
(y -
2
3)
2
2)
=?
=?
(s + 6)2 = ?
11-7
Factor Trinomials
Factoring Pattern for
x2 + bx + c, c positive
x2 + 8x + 15 = (x + 3) (x + 5)
Middle term is the sum of 3 and
5
Last term is the product of 3
and 5
Example
y2 + 14y + 40 = (y + 10) (y + 4)
Middle term is the sum of 10
and 4
Last term is the product of 10
and 4
Example
y2 – 11y + 18 = (y - 2) (y - 9)
Middle term is the sum of -2 and
-9
Last term is the product of -2
and -9
Factoring Pattern for
x2 + bx + c, c negative
x2 - x - 20 = (x + 4) (x - 5)
Middle term is the sum of 4 and
-5
Last term is the product of 4
and - 5
Example
y2 + 6y - 40 = (y + 10) (y - 4)
Middle term is the sum of 10
and -4
Last term is the product of 10
and - 4
Example
y2 – 7y - 18 = (y + 2) (y - 9)
Middle term is the sum of 2 and
-9
Last term is the product of 2
and -9
11-9
More on Factoring
Trinomials
11-9 Factoring Pattern for
ax2 + bx + c
Multiply a(c) = ac
• List the factors of ac
• Identify the factors that add
to b
• Rewrite problem and factor
by grouping
•
Example
2
2x
+ 7x – 9
List factors: (-2)(9) = -18
Factors: (-2)(9) add to 7
(2x2 -2x) + (9x – 9)
2x(x -1) + 9(x – 1)
(x-1)(2x +9)
Example
2
14x
- 17x + 5
List factors: (14)(5) = 70
Factors: (-7)(-10) add to -17
14x2 -7x – 10x + 5
2
(14x – 7x) + (-10x +5)
7x(2x-1)- 5(2x -1)
(7x -5)(2x – 1)
Example
2
3x
- 11x - 4
List factors: (-12)(1) = -12
Factors: (-12)(1) add to -11
3x2 -12x + 1x - 4
2
(3x – 12x) + (1x -4)
3x(x-4) + 1(1x -4)
(x -4)(3x + 1)
END
END
END
END
END