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Section 5.5 – The Real Zeros of a Rational Function

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1 Section 5.5 – The Real Zeros of a Rational Function
Remainder Theorem If f(x) is a polynomial function and is divided by x – c, then the remainder is f(c). Example: 𝑓 π‘₯ = π‘₯ 2 βˆ’2π‘₯βˆ’15 𝐷𝑖𝑣𝑖𝑑𝑒 𝑏𝑦 π‘‘β„Žπ‘’ π‘“π‘Žπ‘π‘‘π‘œπ‘Ÿ: π‘₯βˆ’4 π‘œπ‘Ÿ π‘₯=4 𝑓 4 = 4 2 βˆ’2 4 βˆ’15 𝑓 4 =βˆ’7 The remainder after dividing f(x) by (x – 4) would be -7. 4 8 1 2 βˆ’7

2 Section 5.5 – The Real Zeros of a Rational Function
Factor Theorem If f(x) is a polynomial function, then x – c is a factor of f(x) if and only if f(c) = 0. Example: 𝑓 π‘₯ = π‘₯ 2 βˆ’2π‘₯βˆ’15 𝐷𝑖𝑣𝑖𝑑𝑒 𝑏𝑦 π‘‘β„Žπ‘’ π‘“π‘Žπ‘π‘‘π‘œπ‘Ÿ: π‘₯+3 π‘œπ‘Ÿ π‘₯=βˆ’3 𝑓 βˆ’3 = (βˆ’3) 2 βˆ’2 βˆ’3 βˆ’15 𝑓 βˆ’3 =0 The remainder after dividing f(x) by (x + 3) would be 0. βˆ’3 15 1 βˆ’5

3 Section 5.5 – The Real Zeros of a Rational Function
Rational Zeros Theorem (for functions of degree 1 or higher) Given: (1) 𝑓 π‘₯ = π‘Ž 𝑛 π‘₯ 𝑛 + π‘Ž π‘›βˆ’1 π‘₯ π‘›βˆ’1 + β‹― π‘Ž 1 π‘₯+ π‘Ž 0 (2) Each coefficient is an integer. If 𝑝 π‘ž (in lowest terms) is a rational zero of the function, then p is a factor of π‘Ž 0 and q is a factor of π‘Ž 𝑛 . Theorem: A polynomial function of odd degree with real coefficients has at least one real zero.

4 Section 5.5 – The Real Zeros of a Rational Function
Rational Zeros Theorem Example: Find the solution(s) of the equation. 𝑓 π‘₯ = π‘₯ 3 βˆ’2 π‘₯ 2 βˆ’5π‘₯+6 𝑝: Β±1, Β±2, Β±3, Β± π‘ž: Β±1 𝑝 π‘ž : Β± 1 1 , Β± 2 1 , Β± 3 1 , Β± 6 1 Possible solutions: π‘₯=Β±1, Β±2, Β±3, Β±6 Try: π‘₯= π‘œπ‘Ÿ π‘₯βˆ’1=0

5 Section 5.5 – The Real Zeros of a Rational Function
𝑓 π‘₯ = π‘₯ 3 βˆ’2 π‘₯ 2 βˆ’5π‘₯+6 Long Division Synthetic Division π‘₯ 2 βˆ’π‘₯ βˆ’6 π‘₯ 3 βˆ’π‘₯ 2 1 βˆ’1 βˆ’6 βˆ’π‘₯ 2 βˆ’5π‘₯ 1 βˆ’1 βˆ’6 βˆ’π‘₯ 2 +π‘₯ βˆ’6π‘₯ +6 π‘₯βˆ’1 π‘₯ 2 βˆ’π‘₯βˆ’6 βˆ’6π‘₯ +6 π‘₯βˆ’1 π‘₯ 2 βˆ’π‘₯βˆ’6

6 Section 5.5 – The Real Zeros of a Rational Function
𝑓 π‘₯ = π‘₯ 3 βˆ’2 π‘₯ 2 βˆ’5π‘₯+6 π‘₯βˆ’1 π‘₯ 2 βˆ’π‘₯βˆ’6 =0 π‘₯βˆ’1 π‘₯+2 π‘₯βˆ’3 =0 π‘₯βˆ’1 =0 π‘₯+2 =0 π‘₯βˆ’3 =0 π‘†π‘œπ‘™π‘’π‘‘π‘–π‘œπ‘›π‘ : π‘₯=βˆ’2, 1, 3

7 Section 5.5 – The Real Zeros of a Rational Function
Example: Find the solution(s) of the equation. 𝑓 π‘₯ = 4π‘₯ 4 +7 π‘₯ 2 βˆ’2 𝑝: Β±1, Β± π‘ž: Β±1, Β±2, Β±4 Possible solutions ( 𝑝 π‘ž ): π‘₯=Β±1, Β± 1 2 , Β± 1 4 , Β±2 Try: π‘₯=1 Try: π‘₯=2 4 4 11 11 8 16 46 92 4 4 11 11 9 4 8 23 46 90 Try: π‘₯= 1 2 (π‘₯βˆ’ 1 2 )(4 π‘₯ 3 +2 π‘₯ 2 +8π‘₯+4) 2 1 4 2 4 2 8 4

8 Section 5.5 – The Real Zeros of a Rational Function
(π‘₯βˆ’ 1 2 )(4 π‘₯ 3 +2 π‘₯ 2 +8π‘₯+4) (π‘₯βˆ’ 1 2 )(2)(2 π‘₯ 3 + π‘₯ 2 +4π‘₯+2) (π‘₯βˆ’ 1 2 )(2)( π‘₯ 2 2π‘₯ π‘₯+1 ) (π‘₯βˆ’ 1 2 )(2)( π‘₯ 2 +2) 2π‘₯+1 π‘₯βˆ’ 1 2 = π‘₯ 2 +2= π‘₯+1=0 π‘₯=βˆ’ 1 2 , 1 2 π‘†π‘œπ‘™π‘’π‘‘π‘–π‘œπ‘›π‘ :

9 Section 5.5 – The Real Zeros of a Rational Function
Intermediate Value Theorem In a polynomial function, if a < b and f(a) and f(b) are of opposite signs, then there is at least one real zero between a and b. (𝑏, 𝑓 𝑏 ) (π‘Ž, 𝑓 π‘Ž ) π‘Ÿπ‘’π‘Žπ‘™ π‘§π‘’π‘Ÿπ‘œ π‘Ÿπ‘’π‘Žπ‘™ π‘§π‘’π‘Ÿπ‘œ (𝑏, 𝑓 𝑏 ) (π‘Ž, 𝑓 π‘Ž )

10 Section 5.5 – The Real Zeros of a Rational Function
Intermediate Value Theorem Do the following polynomial functions have at least one real zero in the given interval? 𝑓 π‘₯ =2 π‘₯ 3 βˆ’3 π‘₯ 2 βˆ’2 𝑓 π‘₯ =2 π‘₯ 3 βˆ’3 π‘₯ 2 βˆ’2 [0, 2] [3, 6] 𝑓 0 = βˆ’2 𝑓 2 = 2 𝑓 3 = 25 𝑓 6 = 322 𝑦𝑒𝑠 π‘›π‘œπ‘‘ π‘’π‘›π‘œπ‘’π‘”β„Ž π‘–π‘›π‘“π‘œπ‘Ÿπ‘šπ‘Žπ‘‘π‘–π‘œπ‘› 𝑓 π‘₯ = π‘₯ 4 βˆ’2 π‘₯ 2 βˆ’3π‘₯βˆ’3 𝑓 π‘₯ = π‘₯ 4 βˆ’2 π‘₯ 2 βˆ’3π‘₯βˆ’3 [βˆ’5, βˆ’2] [βˆ’1, 3] 𝑓 βˆ’5 = 587 𝑓 βˆ’2 = 11 𝑓 βˆ’1 = βˆ’1 𝑓 3 = 51 π‘›π‘œπ‘‘ π‘’π‘›π‘œπ‘’π‘”β„Ž π‘–π‘›π‘“π‘œπ‘Ÿπ‘šπ‘Žπ‘‘π‘–π‘œπ‘› 𝑦𝑒𝑠


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