Chabot Mathematics
§5.2 Multiply
PolyNomials
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Review § 5.1
MTH 55
 Any QUESTIONS About
• §5.1 → PolyNomial Functions
 Any QUESTIONS About HomeWork
• §5.1 → HW-15
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Multiply Monomials
 Recall Monomial is a term that is a
product of constants and/or variables
• Examples of monomials: 8, w, 24x3y
 To Multiply Monomials
To find an equivalent expression for the
product of two monomials, multiply the
coefficients and then multiply the
variables using the product rule for
exponents
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
From §1.6  Exponent Properties
a1 = a
0 as an exponent
a0 = 1
Negative exponents a
The Product Rule
The Quotient Rule
1
 n,
a
a n bm  a 
 n,  
m
b
a
b
a m a  a
m
n
a
mn

a
.
n
a
The Power Rule
(am)n = amn
Raising a product
to a power
(ab)n = anbn
Raising a quotient
to a power
Chabot College Mathematics
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n
n
n
a a
   n.
b b
m n
.
n
b
 
a
n
This summary assumes that no
denominators are 0 and that 00 is not
considered. For any integers m and n
1 as an exponent
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  Multiply Monomials
 Multiply:
a) (6x)(7x) b) (5a)(−a)
c) (−8x6)(3x4)
 Solution a) (6x)(7x) = (6  7) (x  x) = 42x2
 Solution b) (5a)(−a) = (5a)(−1a)
= (5)(−1)(a  a) = −5a2
 Solution c) (−8x6)(3x4) = (−8  3) (x6  x4)
= −24x6 + 4 = −24x10
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
(Monomial)•(Polynomial)
 Recall that a polynomial is a monomial
or a sum of monomials.
• Examples of polynomials:
5w + 8, −3x2 + x + 4, x,
0,
75y6
 Product of Monomial & Polynomial
• To multiply a monomial and a polynomial,
multiply each term of the polynomial by
the monomial.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  (mono)•(poly)
 Multiply: a) x & x + 7 b) 6x(x2 − 4x + 5)
 Solution
a) x(x + 7) = x  x + x  7
= x2 + 7x
b) 6x(x2 − 4x + 5) = (6x)(x2) − (6x)(4x) + (6x)(5)
= 6x3 − 24x2 + 30x
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  (mono)•(poly)
 Multiply: 5x2(x3 − 4x2 + 3x − 5)
 Solution:
5x2(x3 − 4x2 + 3x − 5) =
= 5x5 − 20x4 + 15x3 − 25x2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Product of Two Polynomials
 To multiply two polynomials,
P and Q, select one of the
polynomials, say P. Then
multiply each term of P by
every term of Q and combine
like terms.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  (poly)•(poly)
 Multiply x + 3 and x + 5
 Solution
(x + 3)(x + 5) = (x + 3)x + (x + 3)5
= x(x + 3) + 5(x + 3)
=xx+x3+5x+53
= x2 + 3x + 5x + 15
= x2 + 8x + 15
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  (poly)•(poly)
 Multiply
3x − 2 and x − 1
 Solution
(3x − 2)(x − 1) = (3x − 2)x − (3x − 2)1
= x(3x − 2) – 1(3x − 2)
= x  3x − x  2 − 1  3x − 1(−2)
= 3x2 − 2x − 3x + 2
= 3x2 − 5x + 2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  (poly)•(poly)
 Multiply: (5x3 + x2 + 4x)(x2 + 3x)
 Solution:
5x3 + x2 + 4x
x2 + 3x
15x4 + 3x3 + 12x2
5x5 + x4 + 4x3
5x5 + 16x4 + 7x3 + 12x2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  (poly)•(poly)
 Multiply: (−3x2 − 4)(2x2 − 3x + 1)
 Solution
2x2 − 3x + 1
−3x2
−4
−8x2 + 12x − 4
−6x4 + 9x3 − 3x2
−6x4 + 9x3 − 11x2 + 12x − 4
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
PolyNomial Mult. Summary
 Multiplication of
polynomials is an
extension of the
distributive
property. When you
multiply two
polynomials you
distribute each term
of one polynomial to
each term of the
other polynomial.
Chabot College Mathematics
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 We can multiply
polynomials in a
vertical format like
we would multiply
two numbers
(x – 3)
(x – 2)
x_________
–2x + 6
_________
x2 –3x + 0
x2 –5x + 6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
PolyNomial Mult. By FOIL
 FOIL Method
 FOIL Example
(x – 3)(x – 2) = x(x) + x(–2) + (–3)(x) + (–3)(–2) = x2 – 5x + 6
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
FOIL Example
 Multiply (x + 4)(x2 + 3)
FOIL applies to ANY
set of TWO BiNomials,
Regardless of the
BiNomial Degree
 Solution
F O
I
L
(x + 4)(x2 + 3) = x3 + 3x + 4x2 + 12
L
F
I
O
= x3 + 4x2 + 3x + 12
 The terms are rearranged in
descending order for the final answer
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
More FOIL Examples
 Multiply (5t3 + 4t)(2t2 − 1)
 Solution:
(5t3 + 4t)(2t2 − 1) = 10t5 − 5t3 + 8t3 − 4t
= 10t5 + 3t3 − 4t
 Multiply (4 − 3x)(8 − 5x3)
 Solution:
(4 − 3x)(8 − 5x3) = 32 − 20x3 − 24x + 15x4
= 32 − 24x − 20x3 + 15x4
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Special Products
 Some pairs of binomials have special
products (multiplication results).
 When multiplied, these pairs of
binomials always follow the
same pattern.
 By learning to recognize these pairs of
binomials, you can use their
multiplication patterns to find the
product more quickly & easily
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Difference of Two Squares
 One special pair of binomials is the sum
of two numbers times the difference of
the same two numbers.
 Let’s look at the numbers x and 4. The
sum of x and 4 can be written (x + 4).
The difference of x and 4 can be written
(x − 4). The Product by FOIL:
(x + 4)(x – 4) = x2(– 4x + 4x)– 16 = x2 – 16
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Difference of Two Squares
 Some More Examples
}
(x + 4)(x – 4) = x2 – 4x + 4x – 16 = x2 – 16
(x + 3)(x – 3) = x2 – 3x + 3x – 9 = x2 – 9
What do all
of these have
in common?
(5 – y)(5 + y) = 25 +5y – 5y – y2 = 25 – y2
 ALL the Results are Difference of 2-Sqs:
Formula → (A + B)(A – B) = A2 – B2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
General Case F.O.I.L.
 Given the product of generic Linear
Binomials (ax+b)·(cx+d) then FOILing:
Can be Combined IF BiNomials are LINEAR
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Geometry of BiNomial Mult
 The products of
two binomials can
be shown in terms
of geometry; e.g,
(x+7)·(x+5) →
(Length)·(Width)
Length
= (x+7)
x
5
x2
5x
7x
35
Width
= (x+5)
x
7
 (Length)·(Width) = Sum of the areas of
the
four internal
2

x
 5 x  7 x  35
x  7  x  5
rectangles
 x 2  12 x  35
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  Diff of Sqs
 Multiply (x + 8)(x − 8)
 Solution: Recognize from Previous
Discussion that this formula Applies
(A + B)(A − B) = A2 − B2
 So (x + 8)(x − 8) = x2 − 82
= x2 − 64
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  Diff of Sqs
 Multiply (6 + 5w)(6 − 5w)
 Solution: Again Diff of 2-Sqs
Applies → (A + B)(A − B) = A2 − B2
 In this Case
•A6
&
B  5w
 So (6 + 5w) (6 − 5w) = 62 − (5w)2
= 36 − 25w2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Square of a BiNomial
 The square of a binomial is the square
of the first term, plus twice the product
of the two terms, plus the square of
the last term.
These are
called
perfect-square
trinomials
 (A + B)2 = A2 + 2AB + B2
 (A − B)2 = A2 − 2AB + B2
NOTE :
A  B
Chabot College Mathematics
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2
 A B
2
2
A  B
2
 A B
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
2
Example  Sq of BiNomial
 Find: (x + 8)2
 Solution:
Use (A + B)2 = A2+2AB + B2
(x + 8)2 = x2 + 2x8 + 82
= x2 + 16x + 64
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  Sq of BiNomial
 Find: (4x − 3x5)2
 Solution:
Use (A − B)2 = A2 − 2AB + B2
 In this Case
• A  4x
&
B  3x5
(4x − 3x5)2 = (4x)2 − 2  4x  3x5 + (3x5)2
= 16x2 − 24x6 + 9x10
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Summary  Binomial Products
 Useful Formulas for Several Special
Products of Binomials:
For any two numbers A and B,
(A + B)(A − B) = A2 − B2.
For two numbers A and B,
(A + B)2 = A2 + 2AB + B2
For any two numbers A and B,
(A − B)2 = A2 − 2AB + B2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Multiply Two POLYnomials
1. Is the multiplication the product of a
monomial and a polynomial? If so,
multiply each term of the polynomial
by the monomial.
2. Is the multiplication the product of two
binomials? If so:
a) Is the product of the sum and difference
of the same two terms? If so, use pattern
(A + B)(A − B) = A2 − B2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Multiply Two POLYnomials
2. Is the multiplication the product of
Two binomials? If so:
b) Is the product the square of a binomial?
If so, use the pattern
(A + B)2 = A2 + 2AB + B2, or
(A − B)2 = A2 − 2AB + B2
c) c) If neither (a) nor (b) applies, use FOIL
3. Is the multiplication the product of two
polynomials other than those above? If
so, multiply each term of one by every
term of the other (use Vertical form).
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  Multiply PolyNoms
a) (x + 5)(x − 5)
b) (w − 7)(w + 4)
c) (x + 9)(x + 9)
d) 3x2(4x2 + x − 2)
e) (p + 2)(p2 + 3p – 2)

SOLUTION
a) (x + 5)(x − 5) = x2 − 25
b) (w − 7)(w + 4) = w2 + 4w − 7w − 28
= w2 − 3w − 28
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  Multiply PolyNoms

SOLUTION
c) (x + 9)(x + 9)
= x2 + 18x + 81
d) 3x2(4x2 + x − 2) = 12x4 + 3x3 − 6x2
e) By
columns
Chabot College Mathematics
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p2 + 3p − 2
p+2
2p2 + 6p − 4
p3 + 3p2 − 2p
p3 + 5p2 + 4p − 4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Function Notation
 From the viewpoint of functions, if
f(x) = x2 + 6x + 9
and
g(x) = (x + 3)2
 Then for any given input x, the outputs
f(x) and g(x) above are identical.
 We say that f and g represent the
same function
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  f(a + h) − f(a)
 For functions f described by second
degree polynomials, find and simplify
notation like f(a + h) and f(a + h) − f(a)
 Given f(x) = x2 + 3x + 2, find and
simplify f(a+h) and simplify f(a+h) − f(a)
 SOLUTION
f(a + h) = (a + h) 2 + 3(a + h) + 2
= a 2 + 2ah + h 2 + 3a + 3h + 2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  f(a + h) − f(a)
 Given f(x) = x2 + 3x + 2, find and
simplify f(a+h) and simplify f(a+h) − f(a)
 SOLUTION
f (a + h) − f (a) =
[(a + h)2 + 3( a + h) + 2] − [a2 + 3a + 2]
= a 2 + 2ah + h 2 + 3a + 3h + 2 − a2 − 3a − 2
= 2ah + h2 + 3h
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Multiply PolyNomials as Fcns
 Recall from the discussion of the
Algebra of Functions The product of two
functions, f·g, is found by
(f·g)(x) = [f(x)]·[g(x)]
 This can (obviously) be applied to
PolyNomial Functions
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Example  Fcn Multiplication
 Given PolyNomial Functions
f ( x)  x  3
2
g ( x)  x  6 x  8
 Then Find: (f·g)(x) and (f·g)(−3)
 SOLUTION
(f · g)(x) = f(x) ·
g(x)
  x  3 x 2  6 x  8


2
2
(f · g)(−3)
  3  3 3  10  3  24
3
2
 27  27  30  24
 x  6 x  8x  3x  18x  24
0
3
2
 x  3x  10 x  24
3
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
WhiteBoard Work
 Problems From §5.2 Exercise Set
• 30, 54, 82, 98b, 116, 118
 Perfect
Square
Trinomial
By
Geometry
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
All Done for Today
Remember
FOIL By
BIG NOSE
Diagram
Chabot College Mathematics
39
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
Graph y = |x|
6
 Make T-table
x
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Chabot College Mathematics
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5
y = |x |
6
5
4
3
2
1
0
1
2
3
4
5
6
y
4
3
2
1
x
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
file =XY_Plot_0211.xls
-6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt
4
5
6
y
5
4
3
2
1
x
0
-3
-2
-1
0
1
2
3
4
5
-1
-2
M55_§JBerland_Graphs_0806.xls
-3
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt