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Unit F Student Success Sheet (SSS) Finding Zeroes of Polynomials (sections 2.3-‐2.5) Standards: Alg 2 3.0, 5.0, 6.0; Analysis 4.0 Mr. Werdel Segerstrom High School Math Analysis Honors 2011-‐2012 Reminders: • Name: __________________________ Period: _____ • Homework is completed in spiral bound notebook only. Homework not done in homework notebook will not be accepted. All pages in homework notebook should be labeled accordingly: Unit ______ Concept ______ -‐ (title of assignment) Examples: Unit F Concept 1 – Practice Quiz Unit F Concept 1 – Quiz Review Unit F Concept 1-‐4 – Practice Test Need Help? Support is available! nd • Mr. Werdel: Monday after school, 2 lunch. • Mrs. Kirch: Monday – Wednesday Mornings 7-‐8am & Wednesday – Friday afterschool from 3-‐4pm • Ms. Tamaoki: Tuesday & Thursday mornings 7:30-‐8am Success is dependent on effort. Sophocles Concept # What we will be learning… Optional Extra practice from textbook 1 Polynomial long division Worksheet 1 2 Polynomial synthetic division Worksheet 1 3 Using remainder theorem- take one known zero of cubic polynomial to find remaining zeroes Worksheet 2 4 Finding all possible rational zeroes (p's and q's) Worksheet 2 5 Finding possible + and – real zeroes (Descartes) Worksheet 2 6 Given polynomial of 4th or 5th degree, find all zeroes (all real) [utilize Rational Roots Thm, Descartes Rule of Signs] Worksheet 3 7 Adding and subtracting complex numbers Worksheet 4 8 Multiplying and dividing complex numbers Worksheet 4 9 Plotting complex numbers Worksheet 4 10 Given polynomial of 4th or 5th degree, find all zeroes (real and complex zeroes) Worksheet 5 IN THIS UNIT… We will now be looking at polynomials whose parts are not so easy to find through factoring. We will begin by reviewing long division and synthetic division, which are both processes that help us to verify and/or find zeroes. While we will use synthetic division more in this Unit as we look at the remainder theorem and factor theorem, long division will be essential in Unit G. We will be exploring several different ways to go about finding the zeroes of polynomials when factoring is not possible, using such rules as the Rational Roots Theorem (p’s and q’s) and Descartes Rule of Signs. These methods will help us find all of the zeroes, both real and complex of any degree polynomial. -‐-‐-‐Unit F Student Success Sheet-‐-‐-‐Finding Zeroes of Polynomials (sections 2.3-‐2.5)-‐-‐-‐Math Analysis 2011-‐2012-‐-‐-‐ Page 2 #1 Polynomial long division *Make sure polynomial is written in standard form [written with the highest degree first, followed in order counting down] and that If terms are missing, you put a placeholder, such as 0x3 Set up the problem like this • Divide the first term of the dividend by the first term of the divisor, and put that in the answer on top. • Multiply the divisor by that answer, put that below the dividend. Keep all terms lined up by degree • Subtract to create a new polynomial Here are two examples, one with no remainder and one with a remainder. Ex. 1 ( 2x 3 ) ( ) − 5x 2 + x − 10 ÷ x 2 − 4x + 1 1. 2. -‐-‐-‐Unit F Student Success Sheet-‐-‐-‐Finding Zeroes of Polynomials (sections 2.3-‐2.5)-‐-‐-‐Math Analysis 2011-‐2012-‐-‐-‐ Page 3 #2 Polynomial synthetic division Synthetic division works when the divisor is • Binomial (2 terms) • Linear (highest exponent is 1) SYNTHETIC DIVISION ALLOWS US TO DECREASE THE DEGREE OF THE POLYNOMIAL BY ONE! If “m+5” is the divisor, then “-‐5” goes outside the synthetic division bar If “x-‐2” is the divisor, then “2” goes outside the synthetic division bar In synthetic division, we only use the COEFFICIENTS of the polynomial. We still must remember to have a PLACEHOLDER for any missing terms (coefficient of “0”), and to have the polynomial in STANDARD FORM. 1. 2. Bring down the first number Multiply the first # by the divisor and write below the next column 3. Add numbers together 4. Repeat until the end Ex. Write answer as a polynomial of ONE DEGREE LESS than the original. If you have a remainder, put it over the original divisor. 1. 2. 3. -‐-‐-‐Unit F Student Success Sheet-‐-‐-‐Finding Zeroes of Polynomials (sections 2.3-‐2.5)-‐-‐-‐Math Analysis 2011-‐2012-‐-‐-‐ Page 4 #3 Using remainder theorem-‐ take one known zero of cubic polynomial to find remaining zeroes The Remainder Theorem: f ( x ) = 3x 3 − 2x 2 + 5x − 3 You try it! 1. Find f(-‐5) Evaluate f ( 3) 2. Find f(-‐1) Evaluate f ( −1) The Factor Theorem: Ex. Show that ( x − 2 ) and ( x + 3) are factors of f ( x ) = 2x 4 + 7x 3 − 4x 2 − 27x − 18 Use synthetic division to state if the given divisor is a factor. In addition, write the value of the function you find. 1. 2. -‐-‐-‐Unit F Student Success Sheet-‐-‐-‐Finding Zeroes of Polynomials (sections 2.3-‐2.5)-‐-‐-‐Math Analysis 2011-‐2012-‐-‐-‐ Page 5 Once you have found one ZERO, you can use your resulting answer row for the next part of the problem, reduced by one degree! Once you have a quadratic, factor or use the quadratic formula to get the last two zeroes. If x = 3 and x = −1 are zeros of the polynomial f ( x ) = 2x 4 + x 3 − 19x 2 − 9x + 9 , find the other zeros. Use synthetic division to find all of the zeroes of the polynomial, starting with the given factor. Write out the zeroes and the complete factorization. 1. Find all zeros of f ( x ) = x 3 − 11x 2 + 4x + 6 if one zero is x = 1 2. Find all zeros of f ( x ) = 7x 3 + 16x 2 + 11x + 2 if one zero is x = −1 . -‐-‐-‐Unit F Student Success Sheet-‐-‐-‐Finding Zeroes of Polynomials (sections 2.3-‐2.5)-‐-‐-‐Math Analysis 2011-‐2012-‐-‐-‐ Page 6 #4 Finding all possible real zeroes (p's and q's) So… what if I don’t help you out at all? What if I don’t hint to you what a zero might be, and you have to start from scratch RATIONAL ROOTS THEOREM 1. Find the rational zeros of f ( x ) = 2x 3 + 3x 2 − 8x + 3 2. Find the rational zeros of f ( x ) = x 3 + x + 1 p=all the factors of the constant term q=all the factors of the leading coefficient divide p/q to get all the possible rational zeroes, and that’s where you start! If you have a lot to choose from, you can use your nd graphing calculator (zero/root feature under “2 calc” menu) to help you narrow it down! *REMEMBER* -‐These are the possible RATIONAL zeroes; there could be imaginary or irrational (radical) zeroes as well 3. Find the rational zeros of f ( x ) = 2x 4 − 17x 3 + 35x 2 + 9x − 45 4. Find all real zeros of 5. Find all real zeros of f ( x ) = x 3 − 4x 2 − 4x + 16 -‐-‐-‐Unit F Student Success Sheet-‐-‐-‐Finding Zeroes of Polynomials (sections 2.3-‐2.5)-‐-‐-‐Math Analysis 2011-‐2012-‐-‐-‐ Page 7 #5 Finding possible + and – real zeroes (Descartes Rule of Signs) We can also narrow down our possibilities by finding how many POSITIVE answers we could get and how many NEGATIVE answers we could get, based on sign changes… f ( x ) = 3x 4 + 5x 3 − 6x 2 + 8x − 3 1. 2. f(x) Coefficients: + -‐ + -‐ -‐ + f(x) Coefficients: + real zeroes: _4, 2 , or 0__ + real zeroes: _____________ f(-‐x) coefficients: -‐ -‐ -‐ -‐ + + f(-‐x) coefficients: -‐ Real zeroes: _____________ -‐ Real zeroes: __1__ 3. f(x) Coefficients: 4. f(x) Coefficients: + real zeroes: _____________ + real zeroes: _____________ f(-‐x) coefficients: f(-‐x) coefficients: -‐ Real zeroes: _____________ -‐ Real zeroes: _____________ -‐-‐-‐Unit F Student Success Sheet-‐-‐-‐Finding Zeroes of Polynomials (sections 2.3-‐2.5)-‐-‐-‐Math Analysis 2011-‐2012-‐-‐-‐ Page 8 th th #6 Given polynomial of 4 or 5 degree, find all zeroes (all real) [utilize Rational Roots Thm, Descartes Rule of Signs] Let’s put it all together! 1. 2. -‐-‐-‐Unit F Student Success Sheet-‐-‐-‐Finding Zeroes of Polynomials (sections 2.3-‐2.5)-‐-‐-‐Math Analysis 2011-‐2012-‐-‐-‐ Page 9 3. #7 Adding and subtracting complex numbers Complex numbers are in the form a+bi, where “a” is the “real part” and “bi” is the imaginary part To add or subtract, combine all the real parts together and then all of the imaginary parts together! = -‐14 + 6i -‐-‐-‐Unit F Student Success Sheet-‐-‐-‐Finding Zeroes of Polynomials (sections 2.3-‐2.5)-‐-‐-‐Math Analysis 2011-‐2012-‐-‐-‐ Page 10 #8 Multiplying and dividing complex numbers To multiply, use distribution or FOILing to simplify. Remember, i2 = -‐1! 1. 3. 2. 4. ( 6 + 5i ) ( 6 − 5i ) To divide, multiply the top and bottom by the conjugate OF THE DENOMINATOR. Then, use distribution or FOILing to simplify both the top and bottom. Your final denominator must be a REAL NUMBER! (no i) Remember, i2 = -‐1! 5. 6. #9 Plotting complex numbers Complex numbers are plotted just like ordered pairs. Instead of an “x-‐axis”, it is the “real axis”. Instead of the “y-‐axis”, it is the “imaginary axis”. Just like you plot the ordered pair (x,y), you will plot the complex number a+bi as (a,b) -‐-‐-‐Unit F Student Success Sheet-‐-‐-‐Finding Zeroes of Polynomials (sections 2.3-‐2.5)-‐-‐-‐Math Analysis 2011-‐2012-‐-‐-‐ Page 11 1. 2. 3. th th #10 Given polynomial of 4 or 5 degree, find all zeroes (real and complex zeroes) This is the same as concept 6, but now you are not guaranteed that all the zeroes will be REAL (there will be some i’s out there!) Use anything from this chapter to find all real and complex zeroes… have fun! 1. 2. f ( x ) = 4x 5 + 12x 4 − 11x 3 − 42x 2 + 7x + 30
