Polynomial Expressions

Unit 2, Lesson 2

A1.1.1.5.1

Adding and Subtracting Polynomials

• To add and subtract polynomials, simply combine like terms.

o Combine the coefficients.

o DO NOT change the variables and their exponents!!!!

Example A(𝑥3−2𝑥+8 )+(7 𝑥2+2 𝑥−5 )

𝒙𝟑+𝟕𝒙𝟐+𝟑

Example B(7 𝑥2+2 𝑥−5 )+(𝑥3−2𝑥2+4 𝑥+8 )

𝒙𝟑+𝟓𝒙𝟐+𝟔 𝒙+𝟑

Example C(2 𝑦2+6 𝑦−2 )− ( 𝑦2−2 𝑦+7 )

𝒚𝟐+𝟖 𝒚 −𝟗

Example D(2 𝑥3−5𝑥2+3 𝑥−9 )+ (𝑥3+6 𝑥2+11 )

𝟑 𝒙𝟑+𝒙𝟐+𝟑 𝒙+𝟐

Example E(5 𝑥3− 𝑥2+10 𝑥 )− (6 𝑥3+5 𝑥2−7 𝑥 )

−𝒙𝟑−𝟔 𝒙𝟐+𝟏𝟕𝒙

Example F(4 𝑥3−2𝑥2+5 )− (−𝑥3− 𝑥2+4 𝑥−2 )

𝟓 𝒙𝟑−𝒙𝟐−𝟒 𝒙+𝟕

Example G

, , and

𝟐 𝒙𝟑−𝟐 𝒙𝟐+𝒙+𝟓 𝒚+𝒙𝒚

Add the polynomials

Example H

𝟓 𝒙𝟐−𝟗 𝒚+𝟏𝟏𝒙𝒚 −𝟓 𝒚𝟐

Subtract from

Example I

−𝒙𝟐−𝟕 𝒙+𝟏𝟏𝒚+𝟓 𝒚𝟐

Subtract from

Example J

, , and

−𝒙𝟑+𝟔 𝒙𝟐 𝒚+𝟓 𝒙 𝒚𝟐+𝟑 𝒚𝟑

Add the polynomials

Example K

, , and

𝟕 𝒙𝟐 𝒚+𝟑 𝒙 𝒚𝟐+𝟓 𝒙−𝟖−𝟑𝒚+𝟒 𝒙𝒚

Add the polynomials

Multiplying Polynomials

• To multiply polynomials, multiply each and every term in the first polynomial by each and every term of the second polynomial.

• Combine like terms.

1 Term 2 Terms Example A

7 𝑥 (2 𝑥−1 )

𝟏𝟒𝒙𝟐−𝟕𝒙

1 Term 2 Terms Example B

−3 𝑥 (4 𝑥+8 )

−𝟏𝟐𝒙𝟐−𝟐𝟒𝒙

1 Term 2 Terms Example C

5 𝑥𝑦 (2𝑥+3 𝑦−4 )

𝟏𝟎𝒙𝟐𝒚+𝟏𝟓𝒙 𝒚𝟐−𝟐𝟎 𝒙𝒚

2 Terms 2 Terms• The process for multiplying

binomials remains the same. You multiply each term of the first binomial with each term of the second binomial.

• There is a nifty pneumonic to help you remember the procedure.

2 Terms 2 Terms• There is a nifty pneumonic to help you remember the

procedure.

2 Terms 2 Terms Example D

(3 𝑥+2 ) (4 𝑥−5 )

𝟏𝟐𝒙𝟐−𝟕 𝒙−𝟏𝟎

2 Terms 2 Terms Example E

(2 𝑥+1 ) (3 𝑥+7 )

𝟔 𝒙𝟐+𝟏𝟕𝒙+𝟕

2 Terms 2 Terms Example F

(8 𝑥−2 ) (2𝑥−3 )

𝟏𝟔𝒙𝟐−𝟐𝟖𝒙+𝟔

2 Terms 2 Terms Example G

(9 𝑥𝑦−4 ) (5 𝑥+7 )

𝟒𝟓𝒙𝟐𝒚+𝟔𝟑 𝒙𝒚−𝟐𝟎𝒙−𝟐𝟖

Terms Terms• Multiply each and every term in

the first polynomial by each and every term of the second polynomial.

Terms Terms Example H

(𝑥−4 ) (3 𝑥− 𝑦+3 )

𝟑 𝒙𝟐−𝟗 𝒙+𝟒 𝒚 −𝒙𝒚−𝟏𝟐

Terms Terms Example IWhat is the product of

and ?

−𝟔 𝒙𝟐−𝟏𝟖 𝒙−𝟐𝟎 𝒚𝟐+𝟐𝟒𝒚+𝟐𝟑𝒙𝒚

Terms Terms Example JWhat is the product of

and ?

𝟔𝒂𝟐−𝟕𝒂−𝟐𝟎𝒃𝟐+𝟏𝟕𝒃+𝟕𝒂𝒃−𝟑

Terms Terms Example KWhat is the product of

, and ?

𝟏𝟐𝒙𝟑=𝟖 𝒙𝟐−𝟐𝟕𝒙+𝟏𝟖

Example #1

Example #2

Example #3

Example #4

Example #5

Example #6

Example #7

Example #8

Example #9