Drill #29Simplify each expression.
2
3
4
22
4
24
410
324
12432
3
2.5
)3(.4
2
6.3
)3(.2
)2)(3.(1
ab
ba
x
y
yx
yx
ba
dcabcba
Drill #30Simplify each expression.
2
4
3
2321
22542
24
43
3
2
5.5
)6()3(.4
)3)(2(.3
4
10.2
4
3.1
ab
ba
zxyxyz
baba
yx
yx
Drill #20Simplify each expression.
2
42
5
2312
3
4.2
)4()2
1(.1
ba
ab
xyzzxy
)235(2.4
)2
13()4
2
13(.3
2222
2222
xyxyyxxyxyyx
yxyxyxyx
Drill #21Simplify each expression.
)2)(2(.3
)2
13
3
1(2
5
1
6
5.2
)3
2(
4
3.1
2222
2321
32
yxyx
yxyxyxyx
zyxz
xy
Drill #22Simplify each expression.
)32)(7(.3
)2
1
3
1(2
3
2.2
)3
2(
4
3.1
2222
2321
22
xx
yxyxyxyx
zyxz
xy
Drill #23Simplify each expression.
)1)(3)(7(.3
)4
1
2
1
6
1(2
3
2.2
)4(8.1
322232
232122
xxx
yyzyyz
zyxxy
Drill #24Simplify each expression.
22221
2
2
2212
)(:
)2)(1(.3
)52(.2
)6
52
2
1(3.1
xyyxyxB
xx
x
yxxyyxxy
Drill #18
Simplify each expression. State the degree and coefficient of each simplified expression:
)3)(4
1(.3
)4)(3)(2(.2
))(3(.1
2332
3
cbacab
rstrsr
xx
6-1 Operations With Polynomials
Objective: To multiply and divide monomials, to multiply polynomials, and to add and subtract polynomial expressions.
Product of Powers *
For any real number a and integers m and n,
Examples:
nmnm aaa
23535
83535 10101010
aaaa
Quotient of Powers *
For any real number a and integers m and n,
Examples:
nmn
m
aa
a
8)3(53
5
2353
5
101010
10
aaa
a
Power of a Power*
If m and n are integers and a and b are real numbers:
Example:
mnnm aa )(
6)3(232 )( xxx
Power of a Product*
If m and n are integers and a and b are real numbers:
Example:
mmm baab )(
333)( yxxy
Find the value of r
Find the value of r that makes each statement true:
1222
42
24
)(
)(
)(
aa
a
xx
x
yy
r
r
r
Find the value of r *
Find the value of r that makes each statement true:
162.
)(.
)(.
)(.
3
1232
93
2
243
r
r
r
r
D
aa
aC
xx
xB
yyA
MonomialsDefinition: An expression that is 1) a number, 2)
a variable, or 3) the product of one or more numbers or variables.
NOTE: variables must have WHOLE number exponents
Constant: Monomial that contains no variables.Coefficients: The numerical factor of a
monomialDegree: The degree of a monomial is the sum of
the exponents of its variables.
Polynomial*Definition: A monomial, or a sum (or difference)
of monomials.
Terms: The monomials that make up a polynomial
Binomial: A polynomial with 2 unlike terms.Trinomial: A polynomial with 3 unlike terms
Note: The degree of a polynomial is the degree of the monomial with the greatest degree.
Polynomials
Determine whether each of the following is a trinomial or binomial…then state the degree:
yxxyyxEx
yxyxyxEx
xxEx
242
22223
25
3:3
34:2
146:1
Like Terms*
Definition: Monomials that are the same (the same variables to the same power) and differ only in their coefficients.
Examples:
4
3,
3,10
3,23333
5252
abcabc
zyxzyx
yxyx
To combine like terms
To add like terms add the coefficients of both terms together
Example
3
5)3
5()
3
21(
3
2
4)51(5 2222
abcabcabc
abcabc
yxyxyxyx
Adding Polynomials and Subtracting Polynomials
)53()43(:3
)594()876(:2
)132()654(:1
22
22
22
xxxxex
xxxxex
xxxxex
Multiplying a Polynomial by a Monomial
To multiply a polynomial by a monomial:
1. Distribute the monomial to each term in the polynomial.
2. Simplify each term using the rules for monomial multiplication.
)1296(3
4:2
)32(:1
22
2
xxxEx
xxxyEx
FOIL*
Definition: The product of two binomials is the sum of the products of the
F the first terms
O the outside terms
I the inside terms
L the last terms
F O I L
(a + b) (c + d) = ac + ad + bc + bd
The Distributive Method for Multiplying Polynomials*
Definition: Two multiply two binomials, multiply the first polynomial by each term of the second.
(a + b) (c + d) = c ( a + b ) + d ( a + b )
The FOIL Method (for multiplying Polynomials)*
Definition: Two multiply two polynomials, distribute each term in the 1st polynomial to each term in the second.
(a + b) (c + d + e) = (ac + ad + ae) + (bc + bd + be)
The Distributive Method for Multiplying Polynomials*
Definition: Two multiply two polynomials, multiply the first polynomial by each term of the second.
(a + b) (c + d + e) = c ( a + b ) + d ( a + b ) + e ( a + b )
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