mrs. rivas international studies charter school. how we define polynomials polynomial: polynomial:...
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
How we Define Polynomials
Polynomial: is a single term or the sum of two or more terms containing variables with whole-number exponents
How we Describe Polynomials
π ππβπ ππ+πππ βπ
Standard Form
Degree of the
Polynomial
It is customary to write the terms in the order of descending powers of the variable.
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Definition of a Polynomial in x
A polynomial in x is an algebraic expression of the form
The polynomial is of degree n, is the leading coefficient, and is the constant term.
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Adding and Subtracting PolynomialsPolynomials are added and subtracted by combining like terms.
βπ ππ+ππππ=π ππ
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 1 Adding and Subtracting Polynomials
Preform the indicated operations and simplify:
(βπ ππ+π ππβπ π+π )+(ππππ+π ππβπ πβπ )(βπ ππ+π ππβπ π+π )+(ππππ+π ππβπ πβπ )(βπ ππ+ππππ)+(π ππ+π ππ )+(βπ πβπ π )+ (πβπ )
π ππ+π ππβπππβπ
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Multiplying PolynomialsThe product of two polynomials is obtain by using properties of exponents.
(βπ ππ ) (π ππ) ΒΏβππππ
Furthermore we can use the distributive property to multiply a monomial and a polynomial that is not a monomial.
π ππ (π ππβππ+π)
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 2 Multiply a Binomials and a Trinomial
Multiply: (π π+π ) (ππ+π π+π)(π π+π ) (ππ+π π+π)
ΒΏπ π ( ππ+π π+π)+π(ππ+π π+π)ΒΏπ ππ+π+πππ+π+ππ π+πππ+ππ π+ππΒΏπ ππ+π ππ+πππ+πππ+πππ+ππΒΏπ ππ+π ππ+πππ+πππ+πππ+ππΒΏπ ππ+ππ ππ+πππ+ππ
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 3 Using the FOIL Method
Multiply: (ππ+π ) (ππ+π )
(ππ+π ) (ππ+π )
ππππ+ππ+πππ+πππππ+πππ+π
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Special Products Examples
Sum and difference of two terms
Squaring a Binomial
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Special Products Examples
Cubing a Binomial
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 4 Multiply polynomials in Two Variables
Multiply:
(π+π π ) (π πβπ π )A
(π+π π ) (π πβπ π )
πππβπππ+ππππ βππππ
πππ+π ππβππππ
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 4 Multiply polynomials in Two Variables
Multiply:
(π π+ππ )πB
(π π+ππ )π( π¨+π© )π=π¨π+π π¨π©+π©π
ΒΏ (π π)π+πβπ π βπ π+(π π )π
ΒΏππππ+ππππ+ππ π
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 5 Using special Product
Multiply:
(ππ+π+π π )(ππ+πβπ π )A
( π¨+π© ) ( π¨βπ© )=π¨πβπ©π
(ππ+π+π π )(ππ+πβπ π )ΒΏ (π π+π )πβ (π π )π
ΒΏ (π π )π+π [π π βπ ]+ππβ (π π )π( π¨+π© )π=π¨π+π π¨π©+π©π
ΒΏππππ+πππ+ππβππππ
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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 5 Using special Product
Multiply:
(π π+π+π)πB
( π¨+π© )π=π¨π+π π¨π©+π©π
(π π+π+π )πΒΏ (π π+π )π+π [(ππ+π ) βπ ]+ (π )π
ΒΏπ ππ+π ππ +ππ+π π+π π+π
( π¨+π© )π=π¨π+π π¨π©+π©π
ΒΏ (π π )π+π [(π π β π )]+(π )π+π [(π π+π )βπ ]+ (π )π
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Mrs. Rivas
International Studies Charter
School.
1. Preform the indicated operation and simplify
a)
b)
βππ+ππβππβππ
πππβππππβπ πβπ
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Mrs. Rivas
International Studies Charter
School.3. Multiply a)
b)
c)
d)
e)
d)
ππππβππππ+πππβπ
ππππβπππ+ππ
ππππβππππ +π ππ
πππ+ππππ+ππππ
πππ+πππ βππππ+π
πππ+πππ+πππ+ππ+π π+π
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HomeworkMrs. Rivas
International Studies Charter School
Pages: 53-54 # 10, 12, 18, 20, 22, 24, 26, 28, 32, 34, 38, 42, 44, 46, 52, 54, 56, 58, 64, 66, 68, 74, 76, 82