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Warm-Up 2/24 1. 12 πΆ=ππ=6 3 π 6 3 6 B
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Rigor: You will learn how to divide polynomials and use the Remainder and Factor Theorems. Relevance: You will be able to use graphs and equations of polynomial functions to solve real world problems. MA.912. A.2.11
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2-3 The Remainder and Factor Theorems
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Example 1: Use long division to factor polynomial.
6 π₯ 3 β25 π₯ 2 +18π₯+9; π₯β3 6 π₯ 2 β 7π₯ β 3 π₯β3 6 π₯ 3 β25 π₯ 2 +18π₯+9 β6 π₯ π₯ 2 6 π₯ 3 β18 π₯ 2 β7 π₯ 2 +18π₯+9 +7 π₯ 2 β21π₯ β7 π₯ 2 +21π₯ β3π₯+9 +3π₯β9 β3π₯+9 π₯β3 (6 π₯ 2 β7π₯β3) π₯β3 (2π₯β3)(3π₯+1) So there are real zeros at x = 3, , and β
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Example 2: Divide the polynomial.
9 π₯ 3 βπ₯β3; 3π₯+2 3 π₯ 2 β 2π₯ + 1 3π₯+2 9 π₯ 3 +0 π₯ 2 βπ₯β3 β9 π₯ 3 β6 π₯ 2 9 π₯ 3 +6 π₯ 2 β6 π₯ 2 βπ₯β3 β6 π₯ 2 β4π₯ +6 π₯ 2 +4π₯ 3π₯ β3 β3π₯β2 3π₯+2 β5 9 π₯ 3 βπ₯β3 3π₯+2 =3 π₯ 2 β2π₯+1+ β5 3π₯+2 ,π₯β β 2 3 9 π₯ 3 βπ₯β3 3π₯+2 =3 π₯ 2 β2π₯+1β 5 3π₯+2 ,π₯β β 2 3
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Example 3: Divide the polynomial.
2 π₯ 4 β4 π₯ π₯ 2 +3π₯β11; π₯ 2 β2π₯+7 2 π₯ 2 β 1 π₯ 2 β2π₯+7 2 π₯ 4 β4 π₯ π₯ 2 +3π₯β11 β2 π₯ 4 +4 π₯ 3 β14 π₯ 2 2 π₯ 4 β4 π₯ π₯ 2 β π₯ 2 +3π₯β11 + π₯ 2 β2π₯+7 β π₯ 2 +2π₯β7 π₯ β4 2 π₯ 4 β4 π₯ π₯ 2 +3π₯β11 π₯ 2 β2π₯+7 =2 π₯ 2 β1+ π₯β4 π₯ 2 β2π₯+7
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Example 4a: Divide the polynomial using synthetic division.
(2 π₯ 4 β5 π₯ 2 +5π₯β2)Γ· π₯+2 β 2 2 β 5 5 β 2 β β 4 8 β 6 2 2 β 4 3 β 1 2 π₯ 3 β4 π₯ 2 +3π₯β1 2 π₯ 4 β5 π₯ 2 +5π₯β2 π₯+2 =2 π₯ 3 β4 π₯ 2 +3π₯β1
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Example 4b: Divide the polynomial using synthetic division.
(10 π₯ 3 β13 π₯ 2 +5π₯β14)Γ· 2π₯β3 (10 π₯ 3 β13 π₯ 2 +5π₯β14)Γ·2 (2π₯β3)Γ·2 = 5 π₯ 3 β π₯ π₯β7 π₯β 3 2 (10 π₯ 3 β13 π₯ 2 +5π₯β14) (2π₯β3) 3 2 5 β 13 2 5 2 β 7 β 15 2 3 2 6 5 1 4 β 1 5 π₯ 2 +π₯+4β 1 π₯β =5 π₯ 2 +π₯+4β 2 2π₯β3
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Example 6a: Use the Factor Theorem to determine if the binomials are factors of f(x). Write f(x) in factor form if possible. π π₯ =4 π₯ π₯ π₯ 2 β5π₯+3;(π₯β1), π₯+3 1 4 21 25 β 5 3 β 3 4 21 25 β 5 3 β 4 25 50 45 β β 12 β 27 6 β 3 4 25 50 45 48 4 9 β 2 1 π 1 =48, so (π₯β1 ) is not a factor. π β3 =0, so (π₯+3 ) is a factor. π π₯ = π₯+3 (4 π₯ 3 +9 π₯ 2 β2π₯+1)
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Example 6b: Use the Factor Theorem to determine if the binomials are factors of f(x). Write f(x) in factor form if possible. π π₯ =2 π₯ 3 β π₯ 2 β41π₯β20;(π₯+4), π₯β5 β 4 2 β 1 β 41 β 20 β β 8 36 20 π β4 =0, so (π₯+4 ) is a factor. 2 β 9 β 5 5 2 β 9 β 5 β 10 5 π 5 =0, so (π₯β5 ) is a factor. 2 1 π π₯ = π₯+4 (π₯β5)(2π₯+1)
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β1 math! 2-3 Assignment: TX p115, 4-44 EOE
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