Finding Equations of Polynomial Functions with Given Zeros 푛 푛−1 2 Polynomials are functions of general form 푃(푥) = 푎푛푥 + 푎푛−1 푥 + ⋯ + 푎2푥 + 푎1푥 + 푎0 ′ (푛 ∈ 푤ℎ표푙푒 # 푠) Polynomials can also be written in factored form 푃(푥) = 푎(푥 − 푧1)(푥 − 푧2) … (푥 − 푧푖) (푎 ∈ ℝ) Given a list of “zeros”, it is possible to find a polynomial function that has these specific zeros. In fact, there are multiple polynomials that will work. In order to determine an exact polynomial, the “zeros” and a point on the polynomial must be provided. Examples: Practice finding polynomial equations in general form with the given zeros. Find an* equation of a polynomial with the Find the equation of a polynomial with the following two zeros: 푥 = −2, 푥 = 4 following zeroes: 푥 = 0, −√2, √2 that goes through the point (−2, 1). Denote the given zeros as 푧1 푎푛푑 푧2 Denote the given zeros as 푧1, 푧2푎푛푑 푧3 Step 1: Start with the factored form of a polynomial. Step 1: Start with the factored form of a polynomial. 푃(푥) = 푎(푥 − 푧1)(푥 − 푧2) 푃(푥) = 푎(푥 − 푧1)(푥 − 푧2)(푥 − 푧3) Step 2: Insert the given zeros and simplify. Step 2: Insert the given zeros and simplify. 푃(푥) = 푎(푥 − (−2))(푥 − 4) 푃(푥) = 푎(푥 − 0)(푥 − (−√2))(푥 − √2) 푃(푥) = 푎(푥 + 2)(푥 − 4) 푃(푥) = 푎푥(푥 + √2)(푥 − √2) Step 3: Multiply the factored terms together. Step 3: Multiply the factored terms together 푃(푥) = 푎(푥2 − 2푥 − 8) 푃(푥) = 푎(푥3 − 2푥) Step 4: The answer can be left with the generic “푎”, or a value for “푎”can be chosen, Step 4: Insert the given point (−2, 1) to inserted, and distributed. solve for “푎 “. 3 2 1 = 푎[(−2) − 2(−2)] i.e. if 푎 = 1, 푡ℎ푒푛 푃(푥) = 푥 − 2푥 − 8 1 = 푎[−8 + 4] 2 1 = −4푎 i.e. if 푎 = −2, 푡ℎ푒푛 푃(푥) = −2푥 + 4푥 + 16 1 푎 = − 4 Step 5: Insert the value for “푎” into the *Each different choice for “푎” will result in a polynomial, distribute, and re-write the distinct polynomial. Thus, there are an function. infinite number of polynomials with the two 1 3 1 3 1 zeros 푥 = −2 푎푛푑 푥 = 4. 푃(푥) = − (푥 − 2푥) = − 푥 + 푥 4 4 2 Polynomials can have zeros with multiplicities greater than 1. This is easier to see if the Polynomial is written in factored form. Insert Multiplicities of each zero 푚 푛 푝 푃(푥) = 푎(푥 − 푧1) (푥 − 푧2) … (푥 − 푧푖) Insert zeros Multiplicity - The number of times a “zero” is repeated in a polynomial. The multiplicity of each zero is inserted as an exponent of the factor associated with the zero. If the multiplicity is not given for a zero, it is assumed to be 1. Examples: Practice finding polynomial equations with the given zeros and multiplicities. Find an equation of a polynomial with the Find an equation of a polynomial with the given zeroes and associated multiplicities. given zeros and associated multiplicities. Leave the answer in factored form. Expand the answer into general form. Zeros Multiplicity Zeros Multiplicity 푥 = 1 2 Zeros can 푥 = 0 3 푥 = −2 3 be real or 푥 = −1 2 imaginary 푥 = 3 1 푥 = 1 푥 = − 1 Step 1: Write the factored form of the Polynomial. Step 1: Write the factored form of the Polynomial. 푚 푛 푝 푃(푥) = 푎(푥 − 푧1) (푥 − 푧2) (푥 − 푧3) 푚 푛 푝 푃(푥) = 푎(푥 − 푧 ) (푥 − 푧 ) … (푥 − 푧 ) 1 2 푖 Step 2: Insert the given zeros and their Step 2: Insert the given zeros and their corresponding multiplicities. corresponding multiplicities. 2 3 1 푃(푥) = 푎(푥 − 1) (푥 − −2) (푥 − 3) 3 2 1 1 푃(푥) = 푎(푥 − 0) (푥 − −1) (푥 − ) (푥 − −) Step 3: Simplify any algebra if necessary. The Step 3: Simplify any algebra if necessary. answer can be left with the generic “푎”, or a 3 2 specific value for “푎”can be chosen and 푃(푥) = 푎푥 (푥 + 1) (푥 − )(푥 + ) inserted if requested. Step 4: Multiply the factored terms 2 푃(푥) = 푎(푥 − 1)2(푥 + 2)3(푥 − 3) together. Recall that = −1 ! Note the generic “푎” can be used and distributed, or a i.e. let 푎 = 1, 푡ℎ푒푛 2 3 specific value for “푎” can be chosen and 푃(푥) = (푥 − 1) (푥 + 2) (푥 − 3) inserted if requested. let 푎 = −2, 푡ℎ푒푛 7 6 5 4 3 푃(푥) = −2(푥 − 1)2(푥 + 2)3(푥 − 3) 푃(푥) = 푎푥 + 2푎푥 + 2푎푥 + 2푎푥 + 푎푥 Practice Problems: Try these problems on your own! Find an equation of a Polynomial with the given zeros. 1. Zeros: 푥 = −1, −3 Answer: 푓(푥) = 푎(푥2 + 4푥 + 3) 2. Zeros: 푥 = −2, 2, √3, −√3 Answer: 푓(푥) = 푎(푥4 − 7푥2 + 12) 3. Zeros: 푥 = −4, 4 Answer: 푓(푥) = 푎(푥2 + 16) Find the equation of a Polynomial given the following zeros and a point on the Polynomial. 4. Zeros: 푥 = 0, −4 Point: (−3 , 6) Answer: 푓(푥) = −2푥2 − 8푥 1 3 5 5. Zeros: 푥 = −2, 5 Point: (2 , −3) Answer: 푓(푥) = 푥2 − 푥 − 4 4 2 1 1 6. Zeros: 푥 = −4, −1, 1 Point: (2 , 9) Answer: 푓(푥) = 푥3 + 2푥2 − 푥 − 2 2 2 Find an equation of a Polynomial given the following zeros with the listed multiplicities. In each example, set 푎 = 1. 7. Zero: 푥 = 3, 푀푢푙푡푝푙푐푡푦 2 Answer: 푓(푥) = 푥2 − 6푥 + 9 8. Zero: 푥 = 0, 푀푢푙푡푝푙푐푡푦 2 Answer: 푓(푥) = 푥3 + 2푥2 Zero: 푥 = −2, 푀푢푙푡푝푙푐푦 1 9. Zero: 푥 = −1, 푀푢푙푡푝푙푐푡푦 1 Answer: 푓(푥) = 푥4 − 1 Zero: 푥 = 1, 푀푢푙푡푝푙푐푦 1 Zero: 푥 = −, 푀푢푙푡푝푙푐푦 1 Zero: 푥 = −, 푀푢푙푡푝푙푐푦 1 10. Zero: 푥 = −3, 푀푢푙푡푝푙푐푡푦 2 Answer: 푓(푥) = 푥3 + 4푥2 − 3푥 − 18 Zero: 푥 = 2, 푀푢푙푡푝푙푐푦 1 .