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7.1: Finding Rational Solutions of Polynomial Equations
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Rational Root Theorem Used to find all possible rational roots (i.e., zeros). Steps: Find the factors of the constant term (i.e., p) Find the factors of the coefficient of the leading term (i.e., q) Write all factors as Β± Find all possibilities of π π π π = Β±1,Β±2,Β±5,Β±10 Β±1,Β±3,Β±7,Β±21 =Β±π,Β±π,Β±π,Β±ππ,Β± π π ,Β± π π ,Β± π ππ ,Β± π π ,Β± π π ,Β± π ππ ,Β± π π ,Β± π π ,Β± π ππ ,Β± ππ π ,Β± ππ π ,Β± ππ ππ
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Ex #1 Use the Rational Root Theorem to list all possible rational roots for each equation. a) 2 π₯ 3 β π₯ 2 +2π₯+5=0 b) 3 π₯ 3 +7 π₯ 2 +6π₯β8=0 π π = Β±1,Β±5 Β±1,Β±2 =Β±π,Β± π π ,Β±π,Β± π π π π = Β±1,Β±2,Β±4,Β±8 Β±1,Β±3 =Β±π,Β± π π ,Β±π,Β± π π ,Β±π,Β± π π ,Β±π,Β± π π
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Finding Actual Rational Roots
Steps: List all possible rational roots (i.e., π π ) Test each one using synthetic division When you find one that gives you a remainder of 0, you know that itβs a zero. The rest of the entries in the synthetic division tell you the quotient, or remaining factor. Finish factoring what remains to find other roots.
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Ex #2 Use the Rational Root Theorem to list all possible rational roots for the equation. Then find any actual rational roots. (a) π₯ 3 β4 π₯ 2 β7π₯+10=0
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Cont. (b) 2 π₯ 3 β5 π₯ 2 β28π₯+15=0
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(Cont.) On Your Own (c) 3 π₯ 3 β12 π₯ 2 +3π₯+18=0 Hint: Factor out GCF first! Answer: Possible rational roots = Β±1,Β±2,Β±3,Β±6 Β±1 =Β±1,Β±2,Β±3,Β±6 Actual rational roots = 2, β1, 3
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Engineering Application Ex:
A pen company is designing a gift container for their new premium pen. The marketing department has designed a pyramidal box with a rectangular base. The base width is 1 inch shorter than its base length and the height is 3 inches taller than 3 times the base length. The volume of the box must be 6 cubic inches. What are the dimensions of the box?
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Solution:
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On Your Own: A box company is designing a new rectangular gift container. The marketing department has designed a box with a width 2 inches shorter than its length and a height 3 inches taller than its length. The volume of the box must be 56 cubic inches. What are the dimensions of the box?
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Conjugate Root Theorem
If we know a complex number is a root, then its complex conjugate is also a root! Ex: If 2β4π is a root, so is 2+4π Ex: If 2π is a root, so is β2π
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Ex #3 Write a polynomial function with rational coefficients so that it has the roots: (a) β3 and 2π
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(Cont.) On Your Own (b) 1 and βπ
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