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1 Copyright © Cengage Learning. All rights reserved.
5.3 Multiplying Polynomials: Special Products
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What You Will Learn
Find products with monomial multipliers
Multiplying binomials using the Distributive
Property and the FOIL Method
Multiply polynomials using a horizontal or
vertical format
Identify and use special binomial products
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Monomial Multipliers
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Monomial Multipliers
To multiply polynomials, you use many of the rules for
simplifying algebraic expressions.
1. The Distributive Property
2. Combining like terms
3. Removing symbols of grouping
4. Rules of exponents
The simplest type of polynomial multiplication involves a
monomial multiplier.
The product is obtained by direct application of the
Distributive Property.
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Monomial Multipliers
For instance, to multiply the monomial x by the polynomial
(2x + 5), multiply each term of the polynomial by x.
(x)(2x + 5) = (x)(2x) + (x)(5) = 2x2 + 5x
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Example 1 – Finding Products with Monomial Multipliers
Find each product.
a. (3x – 7)(–2x)
b. 3x2(5x – x3 + 2)
c. (–x)(2x2 – 3x)
Solution:
a. (3x – 7)(–2x) = 3x(–2x) – 7(–2x)
= –6x2 + 14x
Distributive Property
Write in standard form.
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cont’d
b. 3x2(5x – x3 + 2)
= (3x2)(5x) – (3x2)(x3) + (3x2)(2)
= 15x3 – 3x5 + 6x2
= –3x5 + 15x3 + 6x2
c. (–x)(2x2 – 3x) = (–x)(2x2) – (–x)(3x)
= –2x3 + 3x2
Distributive Property
Rules of exponents
Write in standard form.
Distributive Property
Write in standard form.
Example 1 – Finding Products with Monomial Multipliers
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Multiplying Binomials
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Multiplying Binomials
To multiply two binomials, you can use both (left and right)
forms of the Distributive Property.
For example, if you treat the binomial (5x + 7) as a single
quantity, you can multiply (3x – 2) by (5x + 7) as follows.
(3x – 2)(5x + 7) = 3x(5x + 7) – 2(5x + 7)
= (3x)(5x) + (3x)(7) – (2)(5x) – 2(7)
= 15x2 + 21x – 10x – 14
= 15x2 + 11x – 14
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Multiplying Binomials
With practice, you should be able to multiply two binomials
without writing out all of the steps above.
In fact, the four products in the boxes above suggest that
you can write the product of two binomials in just one step.
This is called the FOIL Method. Note that the words first,
outer, inner, and last refer to the positions of the terms in
the original product.
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Example 2 – Multiplying Binomials with the Distributive Property
Use the Distributive Property to find each product.
a. (x – 1)(x + 5)
b. (2x + 3) (x – 2)
Solution:
a. (x – 1)(x + 5) = x(x + 5) – 1(x + 5)
= x2 + 5x – x – 5
= x2 + (5x – x) – 5
= x2 + 4x – 5
Right Distributive Property
Left Distributive Property
Group like terms.
Combine like terms.
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cont’d
Example 2 – Multiplying Binomials with the Distributive Property
b. (2x + 3)(x – 2) = 2x(x – 2) + 3(x – 2)
= 2x2 – 4x + 3x – 6
= 2x2 + (–4x + 3x) – 6
= 2x2 – x – 6
Right Distributive Property
Left Distributive Property
Group like terms.
Combine like terms.
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Example 3 – Multiplying Binomials using the FOIL Method
Use the FOIL Method to find each product.
a. (x + 4)(x – 4)
b. (3x + 5)(2x + 1)
Solution:
F O I L
a. (x + 4)(x – 4) = x2 – 4x + 4x – 16
= x2 – 16
Note that the outer and inner products add up to zero.
Combine like terms.
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Example 3 – Multiplying Binomials using the FOIL Method
cont’d
F O I L
b. (3x + 5)(2x + 1) = 6x2 + 3x + 10x + 5
= 6x2 + 13x + 5
Combine like terms.
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Example 4 – A Geometric Model of a Polynomial Product
Use the geometric model to show that
x2 + 3x + 2 = (x + 1)(x + 2)
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Example 4 – A Geometric Model of a Polynomial Product
cont’d
Solution
The left part of the model shows that the sum of the areas
of the six rectangle is
x2 + (x + x + x) + (1 + 1) = x2 + 3x + 2
The right part of the model shows that the area of the
rectangle is
(x + 1)(x + 2) = x2 + 2x + x + 2
= x2 + 3x + 2
So, x2 + 3x + 2 = (x + 1)(x + 2)
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Example 5 – Simplifying a Polynomial Expression
Simplify the expression and write the result in standard form
(4x + 5)2
Solution
(4x + 5)2 = (4x + 5)(4x + 5) Repeated multiplication
= 16x2 + 20x + 20x + 25 Use FOIL Method
= 16x2 + 40x + 25 Combine like terms
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Example 6 – Simplifying a Polynomial Expression
Simplify the expression and write the result in standard form
(3x2 – 2)(4x + 7) – (4x)2
Solution
(3x2 – 2)(4x + 7) – (4x)2 = 12x3 + 21x2 – 8x – 14 – (4x)2 Use FOIL Method
= 12x3 + 21x2 – 8x – 14 – 16x2 Square monomial
= 12x3 + 5x2 – 8x – 14 Combine like terms
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Multiplying Polynomials
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Multiplying Polynomials
The FOIL Method for multiplying two binomials is simply a
device for guaranteeing that each term of one binomial is
multiplied by each term of the other binomial.
(ax + b)(cx + d) = ax(cx) + ax(d) + b(cx) + b(d)
F O I L
This same rule applies to the product of any two
polynomials: each term of one polynomial must be
multiplied by each term of the other polynomial.
This can be accomplished using either a horizontal or a
vertical format.
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Example 7 – Multiplying Polynomials Horizontally
Use a horizontal format to find each product.
a. (x – 4)(x2 – 4x + 2)
b. (2x2 – 7x + 1)(4x + 3)
Solution:
a. (x – 4)(x2 – 4x + 2)
= x(x2 – 4x + 2) – 4(x2 – 4x + 2)
= x3 – 4x2 + 2x – 4x2 + 16x – 8
= x3 – 8x2 + 18x – 8
Combine like terms.
Distributive Property
Distributive Property
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Example 7 – Multiplying Polynomials Horizontally
cont’d
b. (2x2 – 7x + 1)(4x + 3)
= (2x2 – 7x + 1)(4x) + (2x2 – 7x + 1)(3)
= 8x3 – 28x2 + 4x + 6x2 – 21x + 3
= 8x3 – 22x2 – 17x + 3
Combine like terms.
Distributive Property
Distributive Property
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Example 10 – Raising a Polynomial to a Power
Use two steps to expand (x – 3)3
Solution:
Step 1: (x – 3)2 = (x – 3)(x – 3)
= x2 – 3x – 3x + 9
= x2 – 6x + 9
Step 2: (x2 – 6x + 9)(x – 3) = (x2 – 6x + 9)(x) – (x2 – 6x + 9)(3)
= x3 – 6x2 + 9x – 3x2 + 18x – 27
= x3 – 9x2 + 27x – 27
So, (x – 3)3 = x3 – 9x2 + 27x – 27
Combine like terms
Use FOIL Method
Repeated multiplication
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Special Products
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Special Products
Some binomial products, such as those in Example 3(a),
has special forms that occur frequently in algebra.
The product
(x + 4)(x – 4)
is called a product of the sum and difference of two
terms.
With such products, the two middle terms cancel, as
follows.
(x + 4)(x – 4) = x2 – 4x + 4x – 16
= x2 – 16
Sum and difference of two terms
Product has no middle term.
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Special Products
Another common type of product is the square of a
binomial.
(4x + 5)2 = (4x + 5)(4x + 5)
= 16x2 + 20x + 20x + 25
= 16x2 + 40x + 25
Square of a binomial
Use FOIL Method.
Middle term is twice the product
of the terms of the binomial.
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Special Products
In general, when a binomial is squared, the resulting middle
term is always twice the product of the two terms.
(a + b)2 = a2 + 2(ab) + b2
Be sure to include the middle term.
For instance, (a + b)2 is not equal to a2 + b2.
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Special Products
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Example 11 – Finding Special Products
a. (5x – 6)(5x + 6) = (5x)2 – (6) 2 = 25x2 – 36
b. (3x + 7)2 = (3x)2 + 2(3x)(7) + (7)2 = 9x2 + 42x + 14
c. (4x + 9)2 = (4x)2 + 2(4x)(9) + (9)2 = 16x2 + 72x + 81
d. (6 + 5x2)2 = (4)2 – 2(6)(5x2) + (5x2)2
= 36 – 60x2 + (5) 2(x2) 2 = 36 – 60x2 + 25x4
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Example 12 – Finding the Dimensions of a Golf Tee
A landscaper wants to reshape a square tee area for the
ninth hole of a golf course. The new tee area will have one
side 2 feet longer and the adjacent side 6 feet longer than
the original tee. The area of the new tee will be 204 square
feet greater than the area of the original tee. What are the
dimensions of the original tee?
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Solution
Verbal Model:
Labels: Original length = original width = x (feet)
Original area = x2 (square feet)
New length = x + 6 (feet)
New width = x + 2 (feet)
Equation: (x + 6)(x + 2) = x2 + 204 Write equation
x2 + 8x + 12 = x2 + 204 Multiply factors
8x + 12 = 204 Subtract x2 from each side
8x = 192 Subtract 12 from each side
x = 24 Divide each side by 8
cont’d
Example 12 – Finding the Dimensions of a Golf Tee
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Page 244 #’s 1 – 10 down the column
Page 245 #’s 19 & 23
Page 247 #’s 31 – 39 down the column
Page 248 #’s 43 – 47 down the column
Page 249 #’s 55 & 59
Page 251 #’s 77 & 78
Homework:
