Section+2.4

** __2.4 Real Zeros of Polynomials Function__ ** ** __Long Division__ ** Ex. With numbers 15283÷28 = 545 23/28 28 √(15283) −__140__ 128 __−112__ 163 −__140__ 23 With polynomials: put each polynomial in descending order, filling 0 in for any missing terms Ex. (2x^4-x ³ -2)/ (2x ² +x+1) 2x ² +x+1√ (2x^4-x ³ +0x ²+0x-2) - __2x^4+x³ +x²_________ -2x³ -x² +0x __-2x³ -x² - x________ x-2 à degree is smaller than the diviser only can be used when the divisor is in the for x-K 1. The polynomials must be in descending order, fillin0 in for any missing terms __Ex1:__ (2x³ +3x+4)/ (x- 3 ) __3__] 2 0 3 4 the 3 in box is - (-3) __ 0 6 18 63__ 2  6  21  67 (67 is remainder) 2. Fill in x’s starting one degree lower than the divident Answer: 2x ² + 6 x+ 21 + { 67 /(x-3) __Ex.2: (__2x³ -3x ² -5x-12)/(x+1) __-1]__ 2 -3 -5 -12 __ 0 -2 5 0__ 2  -5  0 -12  Anwer: 2 x ² -5 x + { -12 /(x+1)} If a polynomial f(x) has a factor x-K, then the remainder is f(x) A polynomial f(x) has a factor x-K if and only if f(x)=0 Ex: Given f(x)= x ³ -5x ²= 8x-10 Find f (2) clue à divide by x-2 Just plug in 2 in the box for synthesis division __2__] 1 -5 8 -10 __ 0 2 -6 4__ 1 -3 2 -6 Answer: f(2) = -6, x-2 is not a factor of f(x) ** __Rational Zeros Theorem__ ** (Recap) rational- can be written as fraction or a repeating decimal constant term: term with no x variable The possible rational zeros come from the ratio __+__ P/q. Where P is the factors of the constant term, and q is the factors of the leading coefficient. __Ex.1. Find the rational zeros:__ f(x)= x³ – 3x² +1 P:1 P/q :± 1 q:1 f(1): __1__] 1 -3 0 1 f(1)=-1 (remainder), it is not a rational zeros __0 1 -2 -2__ 1 -2 -2  -1
 * __ 2.4 Real Zeros of Polynomials Function  __**
 * __Synthetic Division:__ **
 * __Remainder Theorem__ **** : **
 * __Factor Theorem__ **** : **

f(-1): __-1__] 1 -3 0 1 f(-1)= -3, it is not a rational zero __0 -1 4 -1__ 1 -4 4 -3 Answer; f(x)= x³– 3x² +1 has no rational zeros __Ex.2 Find the rational zeros:__ g(x) = 3x³ +4x²-5x-2 P: 2, 1 P/q: ± 1, 1/3, 2, 2/3 q: 3, 1 __-2__] 3 4 -5 -2 __ 0 -6  4  2__ 3 -2  -1 0 3x² - 2x-1=0  → fill in x’s but starting with one degree lower (3x=1)(x-1)=0 <span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Century Gothic','sans-serif'">→ factor to get zeros x= -1/3 x=1 x=-2 these zeros are rational

** __Example of Home Work Problems__ ** P.223 #1-23 odd Divide f(x) by d(x): //3.f(x)=x// //<span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Century Gothic','sans-serif'">³ //// +4x ////<span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Century Gothic','sans-serif'">² //// + 7x -9 ; d(x)= x +3 // You would use synthetic division to solve, with -3 in the box. The Answer is x^2+x+4 + {-21/(x+3)} // 5. f(x)=x^4- 2x ////<span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Century Gothic','sans-serif'">³ //// + 3x ////<span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Century Gothic','sans-serif'">² //// - 4x+6; d(x)= x ////<span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Century Gothic','sans-serif'">² //// + 2x- 1 // ____________________ x² +2x-1 <span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Century Gothic','sans-serif'">√( x^4 - 2x <span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Century Gothic','sans-serif'">³ + 3x ² - 4x + 6) - __x^4 + 2x³ - 1x² + 0x______ 0 - 4x³ + 4x² - 4x __-  - 4x³ - 8x² + 4x_______ 0 +12x² - 8x + 6 __- 12x² + 24x -12__ 0 +-32x +18 Answer : x² -4x-12 + {(-32x +18)/( x² +2x-1) 23//. X+2 is a factor of 4x^3 +9x^2-3x-10// because the reamainder of f(2) is zero. <span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Century Gothic','sans-serif'; mso-ansi-language: ES"> P.224 #25 f(x)= 2x^3 – 6x^2 – 12x +16 zeros: x=-3, 1 <span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Century Gothic','sans-serif'">Use synthetic division to get quadratic equation to find other zeros for the graph. I started with the number one. <span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Century Gothic','sans-serif'">1] 5 -7 -49 51 __ 0 5 -2 -51__ 5 -2 -51 0 5x^2 – 2x -51 (5x-17)(x+3) 5x-17=0 x+3=0 5x=17 x=3 x=3.4 <span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Century Gothic','sans-serif'">Then write answer as a liner factors f9x)=(x=3)(x-1){x-(17/50}