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.
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.
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
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
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)
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
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
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Special Products Examples
Sum and difference of two terms
Squaring a Binomial
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Special Products Examples
Cubing a Binomial
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
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
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
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
Mrs. Rivas
International Studies Charter
School.
1. Preform the indicated operation and simplify
a)
b)
βππ+ππβππβππ
πππβππππβπ πβπ
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
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