chapter 1.4 part 2 the ring of polynomials.pdf
TRANSCRIPT
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Chapter 1.4Chapter 1.4THE RING OF THE RING OF
POLYNOMIALSPOLYNOMIALSPART 2
Special ProductsSpecial Products
2 2 2Square of a Binomial
2x y x xy y
Example 1.4.10Example 1.4.10
2 2
2 2 2
22 4 2 2 2
Perform the indicated operation.Use special products.
1. 3 6 9
2. 2 3 4 12 9
3. 5 10 25
x x x
a b a ab b
p q r p q p qr r
Special ProductsSpecial Products
3 3 2 2 3
3 3 2 2 3
3 3
3
Cube of a Binom
3
ialx y x x y xy y
x y x x y xy y
3 3 2 2 3
3
3 3 2
3
3 2 2 3
3 2 2 3
1. 2 6 12 8
2. 2 38
3 3
36 54
3
27
3
x x x x
a ba
x y x x y x
a b a
y y
x y x x y x y
b b
y
Special ProductsSpecial Products
2 2 3 3
2 2 3 3
Binomial and Trinomialx y x xy y x y
x y x xy y x y
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Example 1.4.12Example 1.4.12
2 2
2 3
2
3 3
2 2 3 3
3
1. 3 3 9 27
2. 3 2 9 6 4 27 8
x
x y x xy y x y
x y
x x x
y y
x xy y x y
y y
EXTRA ITEMS
3213433322 aaaaaa
333 wyx
11 yxyx
2222 22 babababa
Perform the indicated operation
ANS FactoringFactoring
pA
r p
odolyn
ucto
omial is
f two or
i
f
moit c
re pan be ex
olynomia
facto
ls wi
rable in
thration
pressed as a
al coefici
.ents
R
FactoringFactoring
A factorization is if eachfactor
completeprime facis a tor.
Special ProductsSpecial Products
2 2
2 2 2
2 2 3 3
2
x y x y x y
x y x xy y
x y x xy y x y
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Example 1.4.13Example 1.4.13
4 4
2 24 4 2 2
2 2 2 2
2 2
Factor completely.x y
x y x y
x y x y
x y x y x y
Example 1.4.13Example 1.4.13
2 2
4 4 2 2
But these violate the restrictionshence,
is a complete factorization.
x y x yi x yi
x y x y x y
x y x y x y x y
Example 1.4.14Example 1.4.14
2 2 3 3 4 5 2 2 2 3
3 4 6 2 4 6
2 22 3
2 3 2 3
Factor the following polynomialscompletely.1. 2 1 2
2. 25 25
5
5 5
x y x y x y x y xy x y
x y xd x x y d
x xy d
x xy d xy d
Example 1.4.14Example 1.4.14
22
22
2 2 2 2
2
3. 8 16 4
4. 4 12 9 2 3
5. 5 30 45 5 6 9
5 3
x x x
y y y
y yz z y yz z
y z
Example 1.4.14Example 1.4.14
3 3 3
2
333 3
22
6. 27 33 3 9
7.64 4
4 4 16
b bb b b
y yx x
y xy yx x
Example 1.4.14Example 1.4.14
2
2
2 2
8. 7 12 4 3
9. 427 6
10. 4 4 4
w w w w
x y x y
x y x y
x x y
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Properties of the Set of Properties of the Set of Polynomials with + and Polynomials with + and
1. The sum of two polynomialsis a polynomial.
2. Addition of polynomials isassociative.
3. Is there an additive identity?4. Is there an additive inverse
for each polynomial?
Properties of the Set of Properties of the Set of Polynomials with + and Polynomials with + and
5. Addition of polynomials isCommutative.
The set of polynomials together with+ is an abelian group.
Properties of the Set of Properties of the Set of Polynomials with + and Polynomials with + and
6. The product of two poynomialsis a polynomial.
7. Multiplication of polynomials isAssociative.
8. Multiplication is distributiveover addition of polynomials.
Properties of the Set of Properties of the Set of Polynomials with + and Polynomials with + and
The set of polynomials togetherwith + and is a ring.
Is it a commutative ring?
What is the multiplicative identity?
Is it a field?