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

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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


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