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Example 2.6
Section 2.5 Multiplication of Polynomials and Special Products
Slide 1Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
1. Multiplication of Polynomials2. Special Case Products: Difference of Squares and
Perfect Square Trinomials
Example2.6
Section 2.5 Multiplication of Polynomials and Special Products
Slide 2Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Any Homework Questions?
Example 2.6
Multiplication of Polynomials and Special Products
1. Multiplication of Polynomials
Slide 3Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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.
Example 1 Multiplying Monomials
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Multiply.
a. b. c. 2 7x x
Example 1 Multiplying Monomials
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Multiply.
a. b. 3 2 45 2uv u v 2 3 4 26x y x y
Example 2
Multiplication of Polynomials and Special Products
1. Multiplication of Polynomials
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The distributive property is used to multiply polynomials:
Example 3 Multiplying a Polynomial by a Monomial
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Multiply.
a. b.
ExampleSolution:
4 Multiplying a Polynomial by a Monomial
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a. 2 3 5ab ab a b b. 23 5 3y y y
Example 5
Multiplication of Polynomials and Special Products
1. Multiplication of Polynomials
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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.
ExampleSolution:
5 Multiplying a Polynomial by a Polynomial
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Multiply each term in the first polynomial by each term in the second. That is, apply the distributive property.
Simplify.Combine like terms.
Multiply the polynomials
Example
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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.
Example 6 Multiplying a Polynomial by a Polynomial
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Multiply the polynomials.
Example 7 Multiplying a Polynomial by a Polynomial
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Multiply the polynomials.
2 3 5x x
ExampleSolution:
8 Multiplying a Polynomial by a Polynomial
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Multiply the polynomials. 3 5 2 1x x
Example
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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.
Example 9 Multiplying a Polynomial by a Polynomial
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Multiply the polynomials.
Example
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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.
Example 10 Multiplying a Polynomial by a Polynomial
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Multiply the polynomials. 23 2 9 6 4w w w
Example2.6
Multiplication of Polynomials and Special Products
2. Special Case Products: Difference of Squares andPerfect 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.
Example2.6
Multiplication of Polynomials and Special Products
2. Special Case Products: Difference of Squares andPerfect 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.
Example2.6
Multiplication of Polynomials and Special Products
2. Special Case Products: Difference of Squares andPerfect Square Trinomials
The McGraw-
II. The second special case involves the square of a binomial.
Example2.6
Multiplication of Polynomials and Special Products
2. Special Case Products: Difference of Squares andPerfect Square Trinomials
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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 theterms in the binomial.
Example2.6
Multiplication of Polynomials and Special Products
2. Special Case Products: Difference of Squares andPerfect Square Trinomials
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Note: The expression also expands to a perfect square trinomial, but the middle term will be negative:
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FORMULA Special Case Product Formulas
1.
2.
The product is called a difference of squares.
The product is called a perfect square trinomial.
Example 11 Finding Special Products
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a. b.
Multiply.
Example
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The product can be checked by applying the distributive property or using FOIL .
Example 12 Squaring Binomials
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Square the binomials.
a. b.
Example
Slide 28Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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.
Example2.6
Multiplication of Polynomials and Special Products
Slide 29Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
1. Multiplication of Polynomials2. Special Case Products: Difference of Squares and
Perfect Square Trinomials
Example
Slide 30Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
You Try:
a. b. 3 22 4 3x x x 3 2 53 5t s t s
Example
Slide 31Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
You Try:
a. b. 23 2 4 3x x x 3 2 4 3x x
Example
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You Try:
a. b. 5 7 5 7x x 23 2x
