Example 2.6
Section 2.5 Multiplication of
Polynomials and Special Products
1. Multiplication of Polynomials
2. Special Case Products: Difference of Squares and
Perfect Square Trinomials
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Slide 1
2.6
Example
Section 2.5 Multiplication of
Polynomials and Special Products
Any Homework Questions?
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Slide 2
Example 2.6
Multiplication of Polynomials and
Special Products
1. Multiplication of Polynomials
To multiply monomials, first use the associative and
commutative properties of multiplication to group
coefficients and like bases. Then simplify the result by using
the properties of exponents.
Group coefficients and like bases.
Multiply the coefficients and
add the exponents on x.
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Slide 3
Example 1
Multiplying Monomials
Multiply.
a.  2x 7x
b.
c.
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Example 1
Multiplying Monomials
Multiply.
a.  5uv3  2u 2v 4 
b.
  x y  6 x y 
2
3
4
2
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Example 2
Multiplication of Polynomials and
Special Products
1. Multiplication of Polynomials
The distributive property is used to multiply polynomials:
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Slide 6
Example 3
Multiplying a Polynomial by a
Monomial
Multiply.
a.
b.
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Example 4
Solution:
Multiplying a Polynomial by a
Monomial
a. 2ab 3ab  5a  b
2
3
y
 y  5 3 y 
b. 
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Example 5
Multiplication of Polynomials and
Special Products
1. Multiplication of Polynomials
Next, the distributive property will be used to multiply
polynomials with more than one term.
Note: Using the distributive property results in multiplying
each term of the first polynomial by each term of the
second polynomial.
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Slide 9
Example 5
Solution:
Multiplying a Polynomial by a
Polynomial
Multiply the polynomials
Multiply each term in the first
polynomial by each term in the second.
That is, apply the distributive property.
Simplify.
Combine like terms.
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Example
Notice that the product of two binomials equals
the sum of the products of the First terms, the
Outer terms, the Inner terms, and the Last terms.
The acronym FOIL (First Outer Inner Last) can be
used as a memory device to multiply two
binomials.
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Slide 11
Example 6
Multiplying a Polynomial by a
Polynomial
Multiply the polynomials.
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Example 7
Multiplying a Polynomial by a
Polynomial
Multiply the polynomials.
 2x  3 x  5
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Example 8
Solution:
Multiplying a Polynomial by a
Polynomial
Multiply the polynomials.
 3x  5 2x 1
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Example
It is important to note that the acronym FOIL does not
apply to Examples 9 and 10 because the product does
not involve two binomials.
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Slide 15
Example 9
Multiplying a Polynomial by a
Polynomial
Multiply the polynomials.
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Example
Multiplication of polynomials can be performed
vertically by a process similar to column
multiplication of real numbers. For example,
Note: When multiplying by the column method, it is
important to align like terms vertically before adding
terms.
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Slide 17
Example 10
Multiplying a Polynomial by a
Polynomial
Multiply the polynomials.
2
3
w

2
9
w
 6w  4 


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2.6
Example
Multiplication of Polynomials and
Special Products
2. Special Case Products: Difference of Squares and
Perfect Square Trinomials
In some cases the product of two binomials takes on a
special pattern.
I. The first special case occurs when multiplying the sum
and difference of the same two terms.
2.6
Example
Multiplication of Polynomials and
Special Products
2. Special Case Products: Difference of Squares and
Perfect Square Trinomials
Notice that the middle terms are
opposites. This leaves only the
difference between the square of the
first term and the square of the
second term. For this reason, the
product is called a difference of
squares.
2.6
Example
Multiplication of Polynomials and
Special Products
2. Special Case Products: Difference of Squares and
Perfect Square Trinomials
II. The second special case involves the square of a
binomial.
The McGraw-
2.6
Example
Multiplication of Polynomials and
Special Products
2. Special Case Products: Difference of Squares and
Perfect Square Trinomials
When squaring a binomial, the
product will be a trinomial called
a perfect square trinomial. The
first and third terms are formed
by squaring each term of the
binomial. The middle term
equals twice the product of the
terms in the binomial.
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Slide 22
2.6
Example
Multiplication of Polynomials and
Special Products
2. Special Case Products: Difference of Squares and
Perfect Square Trinomials
Note: The expression
also expands to a perfect
square trinomial, but the middle term will be
negative:
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Slide 23
FORMULA Special Case Product Formulas
1.
The product is called a
difference of squares.
2.
The product is called a
perfect square trinomial.
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Slide 24
Example 11
Finding Special Products
Multiply.
a.
b.
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Example
The product can be checked by applying the
distributive property or using FOIL .
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Slide 26
Example 12
Squaring Binomials
Square the binomials.
a.
b.
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Example
The square of a binomial can be checked by
explicitly writing the product of the two binomials
and applying the distributive property or using
FOIL.
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Slide 28
2.6
Example
Multiplication of Polynomials and
Special Products
1. Multiplication of Polynomials
2. Special Case Products: Difference of Squares and
Perfect Square Trinomials
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Slide 29
Example
You Try:
3
2 5

3
t
s
5
t
a. 
 s 

3
2
b. 2 x x  4 x  3
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
Slide 30
Example
You Try:
2
3
x

2
x
   4 x  3
a. 
b. 3x  2 4x  3
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Slide 31
Example
You Try:
a. 5x  75x  7
b.  3 x  2 
2
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Slide 32