2.5 apply the remainder and factor theorem long division and synthetic division pg. 85

2.5 2.5 Apply the Apply the Remainder Remainder and and Factor Theorem Factor Theorem Long Division Long Division and and Synthetic Division Synthetic Division Pg. 85 Pg. 85

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Page 1: 2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85

2.5 2.5 Apply the Apply the RemainderRemainder and and

Factor TheoremFactor Theorem

Long DivisionLong Division and and Synthetic DivisionSynthetic Division

Pg. 85Pg. 85

Page 2: 2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85

Remainder TheoremRemainder Theorem

If a polynomial P is divided by x-k, then the If a polynomial P is divided by x-k, then the remainder is P(k)remainder is P(k)

(x – k) is a factor of a polynomial P iff P(k) = (x – k) is a factor of a polynomial P iff P(k) = 0.0.

What is the remainder when P(x) = x3 + 2 is divided by x + 1

Is x + 1 a factor of P(x) = x3 + 2 ?

Factor TheoremFactor Theorem

Page 3: 2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85

Dividing PolynomialsDividing Polynomials

Dividing Polynomials to get the Dividing Polynomials to get the factors can be done in one of two factors can be done in one of two waysways

Long Division or Synthetic DivisionLong Division or Synthetic Division

Lets start with Long DivisionLets start with Long Division

Page 4: 2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85

4274 3 3 24 5 2 2x x x x

Page 5: 2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85

3 23 4 12 2x x x x Long DivisionLong Division

Page 6: 2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85

3 2 24 2x x x x Long DivisionLong Division

Page 7: 2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85

3 48 3x x Long DivisionLong Division

Page 8: 2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85

3 24 5 2 2x x x x 3 24 5 2 2x x x x

Long DivisionLong Division Synthetic DivisionSynthetic Division

Page 9: 2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85

3 23 4 12 2x x x x Synthetic DivisionSynthetic Division

Page 10: 2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85

3 2 24 2x x x x Synthetic DivisionSynthetic Division

Can’t be done!!!Can’t be done!!!

Page 11: 2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85

3 48 3x x Synthetic DivisionSynthetic Division

Page 12: 2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85

HomeworkHomeworkPg.Pg. 87,87, 1 – 121 – 12

Pg. 88,Pg. 88, 1 – 81 – 8

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