Presentation is loading. Please wait.

# Section 5.5 โ The Real Zeros of a Rational Function

## Presentation on theme: "Section 5.5 โ The Real Zeros of a Rational Function"โ Presentation transcript:

Section 5.5 โ The Real Zeros of a Rational Function
Remainder Theorem If f(x) is a polynomial function and is divided by x โ c, then the remainder is f(c). Example: ๐ ๐ฅ = ๐ฅ 2 โ2๐ฅโ15 ๐ท๐๐ฃ๐๐๐ ๐๐ฆ ๐กโ๐ ๐๐๐๐ก๐๐: ๐ฅโ4 ๐๐ ๐ฅ=4 ๐ 4 = 4 2 โ2 4 โ15 ๐ 4 =โ7 The remainder after dividing f(x) by (x โ 4) would be -7. 4 8 1 2 โ7

Section 5.5 โ The Real Zeros of a Rational Function
Factor Theorem If f(x) is a polynomial function, then x โ c is a factor of f(x) if and only if f(c) = 0. Example: ๐ ๐ฅ = ๐ฅ 2 โ2๐ฅโ15 ๐ท๐๐ฃ๐๐๐ ๐๐ฆ ๐กโ๐ ๐๐๐๐ก๐๐: ๐ฅ+3 ๐๐ ๐ฅ=โ3 ๐ โ3 = (โ3) 2 โ2 โ3 โ15 ๐ โ3 =0 The remainder after dividing f(x) by (x + 3) would be 0. โ3 15 1 โ5

Section 5.5 โ The Real Zeros of a Rational Function
Rational Zeros Theorem (for functions of degree 1 or higher) Given: (1) ๐ ๐ฅ = ๐ ๐ ๐ฅ ๐ + ๐ ๐โ1 ๐ฅ ๐โ1 + โฏ ๐ 1 ๐ฅ+ ๐ 0 (2) Each coefficient is an integer. If ๐ ๐ (in lowest terms) is a rational zero of the function, then p is a factor of ๐ 0 and q is a factor of ๐ ๐ . Theorem: A polynomial function of odd degree with real coefficients has at least one real zero.

Section 5.5 โ The Real Zeros of a Rational Function
Rational Zeros Theorem Example: Find the solution(s) of the equation. ๐ ๐ฅ = ๐ฅ 3 โ2 ๐ฅ 2 โ5๐ฅ+6 ๐: ยฑ1, ยฑ2, ยฑ3, ยฑ ๐: ยฑ1 ๐ ๐ : ยฑ 1 1 , ยฑ 2 1 , ยฑ 3 1 , ยฑ 6 1 Possible solutions: ๐ฅ=ยฑ1, ยฑ2, ยฑ3, ยฑ6 Try: ๐ฅ= ๐๐ ๐ฅโ1=0

Section 5.5 โ The Real Zeros of a Rational Function
๐ ๐ฅ = ๐ฅ 3 โ2 ๐ฅ 2 โ5๐ฅ+6 Long Division Synthetic Division ๐ฅ 2 โ๐ฅ โ6 ๐ฅ 3 โ๐ฅ 2 1 โ1 โ6 โ๐ฅ 2 โ5๐ฅ 1 โ1 โ6 โ๐ฅ 2 +๐ฅ โ6๐ฅ +6 ๐ฅโ1 ๐ฅ 2 โ๐ฅโ6 โ6๐ฅ +6 ๐ฅโ1 ๐ฅ 2 โ๐ฅโ6

Section 5.5 โ The Real Zeros of a Rational Function
๐ ๐ฅ = ๐ฅ 3 โ2 ๐ฅ 2 โ5๐ฅ+6 ๐ฅโ1 ๐ฅ 2 โ๐ฅโ6 =0 ๐ฅโ1 ๐ฅ+2 ๐ฅโ3 =0 ๐ฅโ1 =0 ๐ฅ+2 =0 ๐ฅโ3 =0 ๐๐๐๐ข๐ก๐๐๐๐ : ๐ฅ=โ2, 1, 3

Section 5.5 โ The Real Zeros of a Rational Function
Example: Find the solution(s) of the equation. ๐ ๐ฅ = 4๐ฅ 4 +7 ๐ฅ 2 โ2 ๐: ยฑ1, ยฑ ๐: ยฑ1, ยฑ2, ยฑ4 Possible solutions ( ๐ ๐ ): ๐ฅ=ยฑ1, ยฑ 1 2 , ยฑ 1 4 , ยฑ2 Try: ๐ฅ=1 Try: ๐ฅ=2 4 4 11 11 8 16 46 92 4 4 11 11 9 4 8 23 46 90 Try: ๐ฅ= 1 2 (๐ฅโ 1 2 )(4 ๐ฅ 3 +2 ๐ฅ 2 +8๐ฅ+4) 2 1 4 2 4 2 8 4

Section 5.5 โ The Real Zeros of a Rational Function
(๐ฅโ 1 2 )(4 ๐ฅ 3 +2 ๐ฅ 2 +8๐ฅ+4) (๐ฅโ 1 2 )(2)(2 ๐ฅ 3 + ๐ฅ 2 +4๐ฅ+2) (๐ฅโ 1 2 )(2)( ๐ฅ 2 2๐ฅ ๐ฅ+1 ) (๐ฅโ 1 2 )(2)( ๐ฅ 2 +2) 2๐ฅ+1 ๐ฅโ 1 2 = ๐ฅ 2 +2= ๐ฅ+1=0 ๐ฅ=โ 1 2 , 1 2 ๐๐๐๐ข๐ก๐๐๐๐ :

Section 5.5 โ 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. (๐, ๐ ๐ ) (๐, ๐ ๐ ) ๐๐๐๐ ๐ง๐๐๐ ๐๐๐๐ ๐ง๐๐๐ (๐, ๐ ๐ ) (๐, ๐ ๐ )

Section 5.5 โ 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? ๐ ๐ฅ =2 ๐ฅ 3 โ3 ๐ฅ 2 โ2 ๐ ๐ฅ =2 ๐ฅ 3 โ3 ๐ฅ 2 โ2 [0, 2] [3, 6] ๐ 0 = โ2 ๐ 2 = 2 ๐ 3 = 25 ๐ 6 = 322 ๐ฆ๐๐  ๐๐๐ก ๐๐๐๐ข๐โ ๐๐๐๐๐๐๐๐ก๐๐๐ ๐ ๐ฅ = ๐ฅ 4 โ2 ๐ฅ 2 โ3๐ฅโ3 ๐ ๐ฅ = ๐ฅ 4 โ2 ๐ฅ 2 โ3๐ฅโ3 [โ5, โ2] [โ1, 3] ๐ โ5 = 587 ๐ โ2 = 11 ๐ โ1 = โ1 ๐ 3 = 51 ๐๐๐ก ๐๐๐๐ข๐โ ๐๐๐๐๐๐๐๐ก๐๐๐ ๐ฆ๐๐

Download ppt "Section 5.5 โ The Real Zeros of a Rational Function"

Similar presentations

Ads by Google