Multiplying Polynomials
After studying this lesson, you will be able to:
 Multiply 2 binomials using the FOIL method
 Multiply any 2 polynomials using the distributive
property.
When we multiply 2 binomials, we can use a special version of
the distributive property called the FOIL method. Here's what
FOIL stands for:

Example: (x  4) (y + 3) 
F irst Terms are
multiplied 
x and y are the First Terms 
O utside Terms are
multiplied 
x and 3 are the Outside Terms 
I nside Terms are
multiplied 
4 and y are the Inside Terms 
L ast Terms are
multiplied 
4 and 3 are the Last Terms 
The FOIL method is the same as distributing twice. The FOIL
method only works when multiplying two binomials.
Example 1
(y + 5) (y + 7)
Since we are multiplying two binomials, let's use the FOIL
Method.
1^{ st}: Multiply the First Terms y times y
2^{ nd}: Multiply the Outside Terms y times 7
3^{ rd}: Multiply the Inside Terms 5 times y
4^{ th}: Multiply the Last Terms 5 times 7
This will give us y^{ 2} + 7y + 5y + 35 which
simplifies to y^{ 2} + 12y + 35
Note: We would get the same answer if we
distributed each term in the first binomial to each term in the
second binomial. That would be the same as using the FOIL Method.
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Example 2
(x + 3)(x  6)
Since we are multiplying two binomials, let's use the FOIL
Method.
1^{ st}: Multiply the First Terms x times x
2^{ nd}: Multiply the Outside Terms x times 6
3^{ rd}: Multiply the Inside Terms 3 times x
4^{ th}: Multiply the Last Terms 3 times 6
This will give us x^{ 2}  6x + 3x  18 which
simplifies to x^{ 2}  3x  18
Example 3
( 3x  5 ) ( 5x + 2 )
Since we are multiplying two binomials, let's use the FOIL
Method.
1^{ st}: Multiply the First Terms 3x times 5x
2^{ nd}: Multiply the Outside Terms 3x times 2
3^{ rd}: Multiply the Inside Terms 5 times 5x
4^{ th}: Multiply the Last Terms 5 times 2
This will give us 15x^{ 2} + 6x  25x  10 which
simplifies to 15x^{ 2}  19x  10
Example 4
( 5a + 1) ( 5a  1 )
Since we are multiplying two binomials, let's use the FOIL
Method.
1^{ st}: Multiply the First Terms 5a times 5a
2^{ nd}: Multiply the Outside Terms 5a times 1
3^{ rd}: Multiply the Inside Terms 1 times 5a
4^{ th}: Multiply the Last Terms 1 times 1
This will give us 25a^{ 2}  5a + 5a 1 which
simplifies to 25a^{ 2}  1
Example 5
(2x  5)(3x^{ 2}  5x + 4)
This time we are multiplying a binomial by a trinomial so we
cannot use the FOIL Method. Instead, we will distribute each term
in the binomial to each term in the trinomial.
1^{ st}: Multiply 2x times the trinomial
2^{ nd}: Multiply 5 times the trinomial
This will give us: 6x^{ 3}  10x^{ 2} + 8x 
15x^{ 2} + 25x  20 which simplifies to 6x^{ 3} 
25x^{ 2} + 33x  20
Example 6
(x  2)(x^{ 2}  x  1)
This time we are multiplying a binomial by a trinomial so we
cannot use the FOIL Method. Instead, we will distribute each term
in the binomial to each term in the trinomial.
1^{ st}: Multiply x times the trinomial
2^{ nd}: Multiply 2 times the trinomial
This will give us: x ^{3}  x^{ 2}  1x  2x^{
2} + 2x + 2 which simplifies to x ^{3}  3x^{ 2}
+ x + 2
