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 − + − 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
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
𝟕 𝒙𝟑−𝟗 𝒙𝟐+𝟏𝟑𝒙 −𝟔
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.
−𝟗 𝒙𝟑+𝟏𝟑𝒙𝟑=𝟒 𝒙𝟑
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 1 Adding and Subtracting Polynomials
Preform the indicated operations and simplify:
(−𝟗 𝒙𝟑+𝟕 𝒙𝟐−𝟓 𝒙+𝟑 )+(𝟏𝟑𝒙𝟑+𝟐 𝒙𝟐−𝟖 𝒙−𝟔 )(−𝟗 𝒙𝟑+𝟕 𝒙𝟐−𝟓 𝒙+𝟑 )+(𝟏𝟑𝒙𝟑+𝟐 𝒙𝟐−𝟖 𝒙−𝟔 )(−𝟗 𝒙𝟑+𝟏𝟑𝒙𝟑)+(𝟕 𝒙𝟐+𝟐 𝒙𝟐 )+(−𝟓 𝒙−𝟖 𝒙 )+ (𝟑−𝟔 )
𝟒 𝒙𝟑+𝟗 𝒙𝟐−𝟏𝟑𝒙−𝟑
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.
(−𝟖 𝒙𝟔 ) (𝟓 𝒙𝟑) ¿−𝟒𝟎𝒙𝟗
Furthermore we can use the distributive property to multiply a monomial and a polynomial that is not a monomial.
𝟑 𝒙𝟒 (𝟐 𝒙𝟑−𝟕𝒙+𝟑)
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 2 Multiply a Binomials and a Trinomial
Multiply: (𝟐 𝒙+𝟑 ) (𝒙𝟐+𝟒 𝒙+𝟔)(𝟐 𝒙+𝟑 ) (𝒙𝟐+𝟒 𝒙+𝟔)
¿𝟐 𝒙 ( 𝒙𝟐+𝟒 𝒙+𝟔)+𝟑(𝒙𝟐+𝟒 𝒙+𝟔)¿𝟐 𝒙𝟏+𝟐+𝟖𝒙𝟏+𝟏+𝟏𝟐 𝒙+𝟑𝒙𝟐+𝟏𝟐 𝒙+𝟏𝟖¿𝟐 𝒙𝟑+𝟖 𝒙𝟐+𝟏𝟐𝒙+𝟑𝒙𝟐+𝟏𝟐𝒙+𝟏𝟖¿𝟐 𝒙𝟑+𝟖 𝒙𝟐+𝟏𝟐𝒙+𝟑𝒙𝟐+𝟏𝟐𝒙+𝟏𝟖¿𝟐 𝒙𝟑+𝟏𝟏 𝒙𝟐+𝟐𝟒𝒙+𝟏𝟖
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 3 Using the FOIL Method
Multiply: (𝟑𝒙+𝟑 ) (𝟓𝒙+𝟑 )
(𝟑𝒙+𝟑 ) (𝟓𝒙+𝟑 )
𝟏𝟓𝒙𝟐+𝟗𝒙+𝟏𝟓𝒙+𝟗𝟏𝟓𝒙𝟐+𝟐𝟒𝒙+𝟗
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:
(𝒙+𝟒 𝒚 ) (𝟑 𝒙−𝟓 𝒚 )A
(𝒙+𝟒 𝒚 ) (𝟑 𝒙−𝟓 𝒚 )
𝟑𝒙𝟐−𝟓𝒙𝒚+𝟏𝟐𝒙𝒚 −𝟐𝟎𝒚𝟐
𝟑𝒙𝟐+𝟕 𝒙𝒚−𝟐𝟎𝒚𝟐
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 4 Multiply polynomials in Two Variables
Multiply:
(𝟓 𝒙+𝟑𝒚 )𝟐B
(𝟓 𝒙+𝟑𝒚 )𝟐( 𝑨+𝑩 )𝟐=𝑨𝟐+𝟐 𝑨𝑩+𝑩𝟐
¿ (𝟓 𝒙)𝟐+𝟐∙𝟓 𝒙 ∙𝟑 𝒚+(𝟑 𝒚 )𝟐
¿𝟐𝟓𝒙𝟐+𝟑𝟎𝒙𝒚+𝟗𝒚 𝟐
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 5 Using special Product
Multiply:
(𝟕𝒙+𝟓+𝟒 𝒚 )(𝟕𝒙+𝟓−𝟒 𝒚 )A
( 𝑨+𝑩 ) ( 𝑨−𝑩 )=𝑨𝟐−𝑩𝟐
(𝟕𝒙+𝟓+𝟒 𝒚 )(𝟕𝒙+𝟓−𝟒 𝒚 )¿ (𝟕 𝒙+𝟓 )𝟐− (𝟒 𝒚 )𝟐
¿ (𝟕 𝒙 )𝟐+𝟐 [𝟕 𝒙 ∙𝟓 ]+𝟓𝟐− (𝟒 𝒚 )𝟐( 𝑨+𝑩 )𝟐=𝑨𝟐+𝟐 𝑨𝑩+𝑩𝟐
¿𝟒𝟗𝒙𝟐+𝟕𝟎𝒙+𝟐𝟓−𝟏𝟔𝒚𝟐
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section P-3
POLYNOMIALS
Example 5 Using special Product
Multiply:
(𝟑 𝒙+𝒚+𝟏)𝟐B
( 𝑨+𝑩 )𝟐=𝑨𝟐+𝟐 𝑨𝑩+𝑩𝟐
(𝟑 𝒙+𝒚+𝟏 )𝟐¿ (𝟑 𝒙+𝒚 )𝟐+𝟐 [(𝟑𝒙+𝒚 ) ∙𝟏 ]+ (𝟏 )𝟐
¿𝟗 𝒙𝟐+𝟔 𝒙𝒚 +𝒚𝟐+𝟔 𝒙+𝟐 𝒚+𝟏
( 𝑨+𝑩 )𝟐=𝑨𝟐+𝟐 𝑨𝑩+𝑩𝟐
¿ (𝟑 𝒙 )𝟐+𝟐 [(𝟑 𝒙 ∙ 𝒚 )]+(𝒚 )𝟐+𝟐 [(𝟑 𝒙+𝒚 )∙𝟏 ]+ (𝟏 )𝟐
Mrs. Rivas
International Studies Charter
School.
1. Preform the indicated operation and simplify
a)
b)
−𝒙𝟑+𝒙𝟐−𝟖𝒙−𝟐𝟎
𝟔𝒙𝟑−𝟏𝟏𝒙𝟐−𝟐 𝒙−𝟖
Mrs. Rivas
International Studies Charter
School.3. Multiply a)
b)
c)
d)
e)
d)
𝟏𝟓𝒙𝟑−𝟑𝟏𝒙𝟐+𝟑𝟎𝒙−𝟖
𝟐𝟖𝒙𝟐−𝟒𝟏𝒙+𝟏𝟓
𝟐𝟏𝒙𝟐−𝟐𝟓𝒙𝒚 +𝟔 𝒚𝟐
𝟒𝒙𝟐+𝟏𝟔𝒙𝒚+𝟏𝟔𝒚𝟐
𝟗𝒙𝟐+𝟏𝟐𝒙 −𝟐𝟓𝒚𝟐+𝟒
𝟒𝒙𝟐+𝟒𝒙𝒚+𝟏𝟐𝒙+𝒚𝟐+𝟔 𝒚+𝟗
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
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