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