Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials...

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Polynomials Polynomials By C. D.Toliver By C. D.Toliver

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Transcript of Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials...

Page 1: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

PolynomialsPolynomialsBy C. D.ToliverBy C. D.Toliver

Page 2: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

PolynomialsPolynomials• An algebraic expression with one or

more terms– Monomials have one term, 3x– Binomials have two terms, 3x

+ 4– Trinomials have three terms, x2

+ 3x + 4

Page 3: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Review:Review: Collecting Like Terms Collecting Like Terms

• You simplify polynomial expressions by collecting like terms

• Like terms have the same variable and the same exponent.2a, 5a, -7a are like terms

2a, 5b, 6c are not like terms a3, a2, a are not like terms• Constants are also like terms 2, 0.5, ¼ are like terms

Page 4: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

ReviewReviewCollecting Like TermsCollecting Like Terms

Example 1. Simplify3x2 + 5x – 2x2 + 3x + 73x2 + 5x – 2x2 + 3x + 71x2 + 8x + 7

Page 5: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Review:Review: Collecting Like Terms Collecting Like Terms

Example 2. Simplify(5a + 7b + 6) + (- 8a - 9b + 5)5a + 7b + 6 + - 8a + - 9b + + 55a + 7b + 6 - 8a - 9b + 5-3a – 2b + 11

Page 6: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Review:Review: Collecting Like Terms Collecting Like Terms

Example 4. Simplify(5a + 7b + 6) – (8a - 6b + 5)5a + 7b + 6 - 8a - -6b -+55a + 7b + 6 - 8a + 6b - 5-3a + 13b + 1

Page 7: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Review:Review: Distributive Property Distributive Property

• We also learned that parenthesis mean to multiply.

• We use the distributive property to multiply polynomials

• The distributive property says: a(b+c) = a(b) + a(c)

Page 8: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Review:Review: Distributive Property Distributive Property

Example 1 Multiply3(x+y) 3(x) + 3(y)3x +3y

Page 9: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Review:Review: Distributive Property Distributive Property

Example 2. Multiply4(2x – 3)4(2x) + 4(-3)8x -12

Page 10: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Review:Review: Distribute and Collect Distribute and Collect

• For more complex expressions you may need to distribute and collect like terms.

• Distribute first• Then collect

Page 11: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Review:Review: Distribute and Collect Distribute and Collect

Example 1. Distribute and Collect6(x + 5) - 2(2x – 8)6(x) +6(5) -2(2x) -2(-8) Distribute6x + 30 – 4x + 16 Collect2x + 46

Page 12: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Review:Review: Distribute and Collect Distribute and Collect

Example 2. Distribute and Collect3(2x - 4) + 7(x – 2) 3(2x) + 3(-4) +7(x) + 7(-2)Distribute6x - 12 + 7x - 14 Collect13x - 26

Page 13: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Review:Review: Distribute and Collect Distribute and Collect

Example 3. Distribute and Collect5(y - 3) + 4(6 - 2y)5(y) +5(-3) +4(6) +4(-2y)Distribute5y - 15 +24 – 8y Collect-3y + 9

Page 14: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply Polynomials• In the previous examples, we were

multiplying polynomials by a monomial, e.g., 3 (x+2)

• 3 is a monomial• x+2 is a polynomial• What happens when you multiply two

polynomials, e.g., (x + 4)(x+2)?

Page 15: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply Polynomials• We will look at three different

methods to multiply polynomials• You may prefer one method over

another• Today we will practice all three

methods

Page 16: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply PolynomialsDistributive MethodDistributive Method

Example 1. Multiply(x + 4)(x + 2)x(x+2) + 4(x+2) Distributex(x) + x(2) + 4(x) +4(2) Distributex2 + 2x + 4x + 8 Collectx2 + 6x + 8

Page 17: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply PolynomialsVertical MethodVertical Method

Example 1. Multiply(x + 4)(x + 2) Rewrite vertically

X + 4X + 22x + 8 Multiply

x2 + 4x Multiply x2 + 6x + 8 Combine

Page 18: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply PolynomialsBox MethodBox Method

Example 1. Multiply(x + 4)(x + 2)=x2 + 6x + 8

x2 2x

4x 8

x +2

x

+4

Page 19: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply PolynomialsDistributive MethodDistributive Method

Example 2. Multiply(x - 3)(x + 5)x(x+5) - 3(x+5) Distributex(x) + x(5) - 3(x) -3(5) Distributex2 + 5x - 3x - 15 Collectx2 + 2x - 15

Page 20: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply PolynomialsVertical MethodVertical Method

Example 2. Multiply(x - 3)(x + 5) Rewrite vertically

X - 3X + 5

5x - 15 Multiply x2 - 3x Multiply x2 + 2x - 15 Combine

Page 21: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply PolynomialsBox MethodBox Method

Example 2. Multiply(x - 3)(x + 5)=x2 + 2x - 15

x2 -3x

5x -15

x -3

x

+5

Page 22: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply PolynomialsDistributive MethodDistributive Method

Example 3. Multiply(2x + 1)(x - 4)2x(x-4) + 1(x-4) Distribute2x(x)+2x(-4)+1(x)+1(-4) Distribute2x2 - 8x + 1x -4 Collect2x2 - 7x - 4

Page 23: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply PolynomialsVertical MethodVertical Method

Example 3. Multiply(2x + 1)(x - 4) Rewrite vertically

2x + 1x - 4

-8x - 4 Multiply 2x2 + 1x Multiply 2x2 - 7x - 4 Combine

Page 24: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply PolynomialsBox MethodBox Method

Example 3. Multiply(2x + 1)(x - 4)=2x2 - 7x - 4

2x2 1x

-8x -4

2x +1

x

-4

Page 25: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply PolynomialsDistributive MethodDistributive Method

Example 4. Multiply(2x + 3)(3x - 4)2x(3x-4)+3(3x-4) Distribute2x(3x)+2x(-4)+3(3x)+3(-4)Distribute6x2 - 8x + 9x - 12 Collect6x2 + 1x - 12

Page 26: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply PolynomialsVertical MethodVertical Method

Example 4. Multiply(2x + 3)(3x - 4) Rewrite vertically

2X + 3 3X - 4 -8x - 12 Multiply

6x2 + 9x Multiply 6x2 + 1x - 12 Combine

Page 27: Polynomials By C. D.Toliver. Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4.

Multiply PolynomialsMultiply PolynomialsBox MethodBox Method

Example 4. Multiply(2x + 3)(3x - 4)=6x2 + 1x - 12

6x2 9x

-8x -12

2x +3

3x

-4


X + 5 4x +20x R + 3 3x - 16 3x - +15 -31 - - -. Tues 11/24 Lesson 5 – 4 Learning Objective: To divide polynomials by synthetic division Hw: Pg. 308 #21-29.

X + 5 4x +20x R + 3 3x - 16 3x - +15 -31 - - -. Tues 11/24 Lesson 5 – 4 Learning Objective: To divide polynomials by synthetic division Hw: Pg. 308 #21-29.

Name Adding and Subtracting Polynomials - Mr. Beals …mrbealsquest.weebly.com/uploads/9/6/0/0/9600167/alg2-2.pdf · Name Adding and Subtracting Polynomials + 7x-f)-(x2-3x + 4) =

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Web viewQuadratics: Factoring Polynomials. Name:_____ Steps: GCF – Any number of terms. Four Terms – Grouping. Three Terms – Trinomial Rules. Two Terms

Web viewQuadratics: Factoring Polynomials. Name:_____ Steps: GCF – Any number of terms. Four Terms – Grouping. Three Terms – Trinomial Rules. Two Terms

Let’s Get Together! Combining Like Terms in Polynomials

Let’s Get Together! Combining Like Terms in Polynomials

Adding and Substracting Polynomials · Adding and Subtracting Polynomials What are like terms?Like terms have the same variables with exactly the same exponents Which of the following

Adding and Substracting Polynomials · Adding and Subtracting Polynomials What are like terms?Like terms have the same variables with exactly the same exponents Which of the following

Painlevé equations and orthogonal polynomialsare in terms ofparabolic cylinder functions. Walter Van Assche Painlev e equations and orthogonal polynomials Orthogonal polynomials on

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Multiplying Polynomials Use the distributive property, and remember your properties for exponents. 5x (4x 2 + 3x) = 20x 3 + 15x 2 Section 10.2.

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Chapter 6 Polynomials. 6.1 Adding Polynomials Monomial – one term expression Binomial – two term expression…. Polynomial – “many terms” What.

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Factoring Polynomials Factoring Polynomials Factoring Polynomials.

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Kinds Of Polynomials According To Terms ezdock

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CP Algebra 2 Unit 3A: Polynomials · 3 Polynomials: The Basics After this lesson and practice, I will be able to … ¨ classify polynomials by degree and number of terms. (LT 1)

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