Math 030 - Cooley
Intermediate Algebra
OCC
Section 5.2 – Multiplication of Polynomials
Use the “Distributive Property (of Multiplication over Addition)”
The Distributive Property helps us explain what happens when multiplication is interpreted as repeated
addition. It is the most commonly used technique for multiplying two polynomials.
 Examples:
Monomial  Binomial
2 x  3 x  4    2 x  3 x    2 x  4 

Monomial  Trinomial

 6 x  8x
2
 12 x 2  3x  6
Binomial  Trinomial
Binomial  Binomial
 2 x  3 5x  9   2 x  5x  9   3  5x  9 
 3x  2   2 x 2  5 x  4   3x  2 x 2  5 x  4   2  2 x 2  5 x  4 
 10 x 2  18 x  15 x  27
 6 x3  15 x 2  12 x  4 x 2  10 x  8
 10 x  3x  27
 6 x3  19 x 2  22 x  8
2
 4x
2
 
3 4 x  x  2   3 4 x 2   3 x    3 2 
2


Trinomial  Trinomial

 
 
 x  2 2 x  5x  4  4 x2 2 x2  5x  4  x 2 x2  5x  4  2 2 x2  5x  4
2

 8 x 4  20 x3  16 x 2  2 x3  5 x 2  4 x  4 x 2  10 x  8
 8 x 4  18 x3  7 x 2  14 x  8
Use the “FOIL Method” (First–Outside–Inside–Last)
The “FOIL Method” is an acronym for the four products that are produced when a binomial multiplies a
binomial in the distributive process. It is only used to simplify the product of two binomials. It is not a new
method of multiplication, but rather a summary of the distributive process designed to help students eliminate
some of the scratch work.
The FOIL Method
To multiply two binomials, A + B and C + D, multiply the First terms AC, the Outer terms AD, the Inner terms
BC, and then the Last terms BD. Then combine like terms, if possible.
( A  B)(C  D)  AC  AD  BC  BD
1. Multiply First terms: AC.
2. Multiply Outer terms: AD.
3. Multiply Inner terms: BC.
4. Multiply Last terms: BD.
 Examples:
 2 x  3 5 x  9   10Fx 2  18O x 15I x 27
L
 10 x  3x  27
2
F
O I
L
 3x  y  2 x  3 y   6Fx 2  9 xy  2 xy  3 y 2
O
I
L
 6 x  11xy  3 y
2
F
O I
2
L
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Math 030 - Cooley
Intermediate Algebra
OCC
Section 5.2 – Multiplication of Polynomials
Use “Special Products” or Formulas
“Special Products” or Formulas are used to help us simplify some commonly encountered mathematical structures involved
with polynomial multiplication. The formulas represent what happens after the polynomial multiplication process takes
place and like terms are collected.
The Square of a Binomial
(a.k.a. Binomial Squared)
 a  b   a  2ab  b
2
 a  b   a 2  2ab  b2
2
2
2
Why it Works:
a  b
2
 a  b
2
  a  b  a  b   a 2  ab ab b 2  a 2  2ab b 2
F
I
L
  a  b  a  b   a  ab ab b 2  a 2  2ab b 2
F
O
I
L
 Examples:
 3x  7 y    3x   2  3x  7 y    7 y   9 x 2  42 xy  49 y 2
2
2
2
 2a  3   2a   2  2a  3   3  4a 2  12a  9
2
(These result is a perfect-square trinomial).
The Product of a Sum and Difference
of Two Terms
(a.k.a. Product of Two Conjugates)
 a  b  a  b   a
O
2
2
2
Why it Works:
 ab  b2  a 2  b2
 a  b  a  b   aF2  ab
O
I
L
 Examples:
2
b
 3x  4 y  3x  4 y    3x    4 y 
2
2
2
 9 x 2  16 y 2
(This results in a Difference of Two Squares).
 Exercises:
Multiply, and if possible, simplify
1)
6(5 x 4 )
2)
2 x3  4 x
3)
3uv 2 (5u 2v7 )
4)
3x(x  5)
5)
( x  7)( x  7)
6)
( x  3)( x  3)
7)
(3  x)(6  2 x)
8)
 x  12   x  15 
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Math 030 - Cooley
Intermediate Algebra
OCC
Section 5.2 – Multiplication of Polynomials
 Exercises:
Multiply, and if possible, simplify
9)
( x 2  3)( x  1)
10)
(4 x  1)(2 x  7)
11)
(c  9)2
12)
( y  4)( y  4)
13)
(5t  3)(5t  3)
14)
(3a  2)(3a  2)
15)
( x  5)2
16)
(5  2t 3 ) 2
17)
(3 p 2  p)(3 p 2  p)
18)
(5 x 2  2)(3x 3  8)
19)
( x  2)( x 2  x  7)
20)
( x 2  5 x  1)( x 2  x  3)
21)
(4 x 2  5x  3)( x 2  2 x  4)
22)
(t  5)2  (t  4)(t  4)
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