Powers And Roots Of Complex Numbers Worksheet

Here r = jzj. They include numbers of the form a + bi where a and b are real numbers. For instance, the quadratic equation. Subtracting Integers Using a Number Line. Examples (2+3i)+(5−6i) = 7−3i. Here I give the formula to find the n-th root of a complex number and use it to find the square roots of a number. Complex Numbers and RootsComplex Numbers and Roots Plainlocal. There are other points to consider. The Origin of Complex Numbers. Thus, where other word lists of the Phrontistery are simply listed in "word: definition" form, this page will try to show you, in tabular format, how to construct your own terms from the basic. Z A iA plmlC vr5i0gAhat Js n xr Geus1e0rkv Jeyd 0. The basis of this notion is mainly related to numbers multiplied by the same numbers. a pair of complex numbers that have real parts that are the sa… a number in the form a+bi, where a is the real part, b is the… the square root of any negative number, which has the form bi,…. Doing so helps us to organize our information. conjugates. Then by De Moivre's Formula for the Polar Representation of Powers of Complex Numbers we have that: (2) \begin{align} \quad z^n = r^n (\cos n\theta + i \sin n \theta) \end{align}. Our fraction calculator displays the result in a clear way. Courtesy of Harold Hiken. This number can’t be described as solely real or solely imaginary — hence the term complex. Displaying all worksheets related to - Imaginary Numbers. Complex Numbers The square root of negative one, which we now call the imaginary unit, was once thought to be an absurd notion. The star Sirius is approximately 75 684 000 000 000 km from the sun. If discriminant is greater than 0, the roots are real and different. Roots Of Complex Numbers. † One can view a complex number z = a + bi in polar, just as in Calc 3. We may add it, subtract it, multiply it, and so on. Some examples are shown in Figure 1. It states that, for a positive integer n, z n is found by raising the modulus to the n th power and multiplying the argument by n. 1 Complex Numbers 1. On this page you will find: a complete list of all of our math worksheets relating to complex numbers. Complex numbers make 2D analytic geometry significantly simpler. We hope you enjoy your adventures with complex numbers and they give you a taste for the exciting mathematics you can discover by choosing Further Mathematics at A-level. Using Absolute Value to Combine Integers. But, the question remains, can all polynomial equations be solved using complex numbers? Fundamental Theorem of Algebra: If ( )Pxis a polynomial of degree n 1 with complex coefficients, then Px() 0 has at least one complex root. On this page, you'll find an unlimited supply of printable worksheets for square roots, including worksheets for square roots only (grade 7) or worksheets with square roots and other operations (grades 8-10). Complex numbers is vital in high school math. Gauss used the term \complex" in the early. The Exponents and Radicals Worksheets are randomly created and will never repeat so you have an endless supply of quality Exponents and Radicals Worksheets to use in the classroom or at home. 5) Practice Problems on Powers and Roots. multiplicationanddivision. Complex Numbers Foldable Posted on December 16, 2015 by I Speak Math Complex numbers are amazingly fun to teach in Algebra 2 because it is the first time students have ever seen them. Moivre's formula. For every even-degree root (for example the 2nd, 4th, 6th ) there are two roots. Here are the square roots of all the perfect squares from 1 to 100. This library offers a comprehensive set of functions and is easily extendible. Time to mix it up! We have included a mixture of multiple operations worksheets for many different levels. You can manipulate complex numbers arithmetically just like real numbers to carry out operations. The conjugate of the complex number a + bi is a – bi. In order to use DeMoivre's Theorem to find complex number roots we should have an understanding of the trigonometric form of complex numbers. Algebra with complex numbers. We will also look at some basic properties of each of these and how we can use these properties to work with and simplify. But, the question remains, can all polynomial equations be solved using complex numbers? Fundamental Theorem of Algebra: If ( )Pxis a polynomial of degree n 1 with complex coefficients, then Px() 0 has at least one complex root. As another example, consider 35. Roots and rational exponents. First, z 3 = 1331 z = [r, θ ] by De Moivre’s. This library offers a comprehensive set of functions and is easily extendible. Use the definition 1 i to simplify square roots. Complex Numbers in Standard Form 46 min 12 Examples Intro to Video: Complex Numbers in Standard Form Overview of Real Numbers and Imaginary Numbers Complex Numbers in Standard Form and Addition and Subtraction of Complex Numbers Examples #1-6: Add or Subtract the Complex Numbers and Sketch on Complex Plane Two Examples with Multiplication and Division…. to the nth power is 1 (unity) Many different complex numbers when raised to the nth power can give 1 But this specific one is conventionally called the nth root of unity exp 2 / in exp exp exp 2 1 22 iin ni nn n 1exp2 / in. For example, complex number A + Bi is consisted of the real part A and the imaginary part B, where A and B are positive real numbers. best geometry book. Real numbers = rational numbers + irrational numbers Observation #9: The difference between complex numbers and real numbers is that complex numbers give solutions for the following expressions and more! √(-7), √(1-8), √(-25) = 5i, etc Among the different types of numbers, fractions is among the toughest to understand. Evaluate the square root of negative numbers. Each of the functions including full description, examples of how to use, related tips & tricks and common errors. The obvious identity p 1 = p 1 can be rewritten as r 1 1 = r 1 1: Distributing the square root, we get. SAT Math Help » Algebra » Exponents » Squaring / Square Roots / Radicals » Complex Numbers Example Question #1 : Complex Numbers From , subtract its complex conjugate. Math Robot Project. ExploreMath. Functions can be used to create formulas that manipulate data and calculate strings and numbers. 4, you learned how to add, subtract, multiply, and divide complex numbers. Introduction to Absolute Value. Roots and radicals are the inverse of exponents. Wed Jan 29 Roots of a Complex Numbers: Class 5. org 5-5 Complex Numbers and Roots Every complex number has a real part a and an imaginary part b. Solutions follow the problems. Courtesy of Harold Hiken. The 3 is called the index. Worksheet Solving Quadratic Inequalities and Quadratic Word Problems. Show that the complex number 1+iis a root of the cubic equation z3 +z2 +(5−7i)z−(10+2i)=0, and hence find the other two roots. For which values of a does the polynomial have two distinct real. To complete the basic arithmetic of complex numbers we shall look at determining powers and roots. Square Roots of Complex Numbers. 81 Roots and Powers of Complex Numbers Polar Worksheet 9. Power, Polynomial, and Rational Functions Graphs, real zeros, and end behavior Dividing polynomial functions The Remainder Theorem and bounds of real zeros Writing polynomial functions and conjugate roots Complex zeros & Fundamental Theorem of Algebra Graphs of rational functions Rational equations Polynomial inequalities Rational inequalities. Dividing complex numbers: 1. Apply the Pythagorean Theorem to find the hypotenuse of a right triangle. Write the 3 cube roots of 1331 in trigonometric form. general math worksheets. Students calculate powers of numbers without a calculator. Excel Format. Simplify the radical expression. The expression under the radical sign is called the radicand. In this lesson the learner will be acquinted with complex numbers, its representation and algebraic operations on complex numbers. Find roots of complex numbers in polar form. The polar form is where a complex number is denoted by the length (otherwise known as the magnitude, absolute value, or modulus) and the angle of its vector (usually denoted by an angle symbol that looks like this: ∠). Eighth graders determine the square and cube roots of given numbers in this math lesson. Square roots is a specialized form of our common roots calculator. 1 Complex Numbers 1. In other words, there exists a complex number csuch that P(c) = 0. In general: To simplify a surd, write the number under the root sign as the product of two factors, one of which is the largest perfect square. Complex numbers is vital in high school math. 25 25 1 Factor out 1. 1 Complex numbers: algebra. Determine the conjugate of the denominator. This collection gives you an opportunity to explore these ideas yourself, and discover more about the impact and applications of complex numbers in our everyday lives. Complex numbers can be multiplied and divided. You will find addition lessons, worksheets, homework, and quizzes in each section. The nth root of a complex number z is given by: For example to calculate the 3rd roots of -1 + i, I would write: The last two expressions can be approximated by calculator if necessary. † Geometrically, addition is just as vector addition from Calc 3 (tip-to-tail). The modulus of a complex number is the distance from the origin on the complex plane. In this equation, the highest power of x is 2. Algebra 2/Trig R Homework Worksheets; Complex Roots, Review. a problem. Gauss used the term \complex" in the early. (With and being real numbers and being the imaginary unit). I L RMEa 8d 8eJ pw ei nt Fhx ZIOnWfyiwn BiAtAe Y AXlbgke bEr 0ax c2 i. ) of negative numbers (they are complex numbers), but there are roots of negative numbers if the degree is an uneven number. COMPLEX NUMBERS 5. This collection gives you an opportunity to explore these ideas yourself, and discover more about the impact and applications of complex numbers in our everyday lives. You might notice that all three roots are equally spaced around the full 2*pi circle. The number ais called the real part of. y| = √ x 2 + y 2 is trigonometric form of a complex number (more information on trigonometric functions is here). First, z 3 = 1331 z = [r, θ ] by De Moivre’s. Next we consider powers of i. Cubes and Cube Roots (A) Welcome to The Cubes and Cube Roots (A) Math Worksheet from the Number Sense Worksheets Page at Math-Drills. Moivre's formula. Create a Polynomial Algebra Worksheet This page will create a practice worksheet for you, dealing with polynomials. of each other. That number is the square root of [latex]−1,\sqrt{-1}[/latex]. Choose a specific addition topic below to view all of our worksheets in that content area. The real and imaginary components. Do you notice any relationship between the polar forms of two complex numbers and the polar form of their product? Polar Form Polar Form Polar Form Polar Form. Free practice questions for Precalculus - Powers and Roots of Complex Numbers. A complex number is a number with a Real part, a, and an imaginary part, bi written in the form I. multiplicationanddivision. Learn math online and take free tests. For example, complex number A + Bi is consisted of the real part A and the imaginary part B, where A and B are positive real numbers. The complex number calculator allows to multiply complex numbers online, the multiplication of complex numbers online applies to the algebraic form of complex numbers, to calculate the product of complex numbers `1+i` et `4+2*i`, enter complex_number((1+i)*(4+2*i)), after calculation, the result `2+6*i` is returned. Plane Isometries As Complex Functions; Remarks on the History of Complex Numbers. Numbers such as these are called imaginary numbers. This page contains sites relating to Exponents/Roots. Imaginary numbers have been a bee in my bonnet for years — the lack of an intuitive. 8 Modeling with Quadratic Functions. Create a Polynomial Algebra Worksheet This page will create a practice worksheet for you, dealing with polynomials. You may select the type of problems to use, and this worksheet produces 12 problems per page. 3 Imaginary Numbers Worksheet Templates are collected for any of your needs. DefinitionofImaginaryNumbers: i2 = − 1(thus i = − 1 √) Examples of imaginary numbers include 3i, − 6i, 3 5 i and 3i 5 √. and so forth. Square Roots, Powers & Roots of Complex Numbers - Chapter Summary. Finding Powers of Complex Numbers in Polar Form. See more ideas about Complex numbers, Algebra 2 and Algebra. A range of resources covering all aspects of indices up to GCSE. P(z) is a nonconstant polynomial, then P(z) has a complex root. A given quadratic equation ax 2 + bx + c = 0 in which b 2 -4 ac < 0 has two complex roots: x = ,. Thus, the equation has two roots. This is a 7 th grade topic that falls under algebra ii. New Maths Curriculum Collection 2: Integers, Powers and Roots October 18, 2014 October 18, 2014 Craig Barton Each fortnight, I will put together a selection of resources designed to help cover the objectives of the new maths curriculum for secondary schools (Key Stage 3). ) So if we want to find the four fourth roots of 16 we solve this equation. From this, it is obvious that two complex numbers (a+bi) and (c+di) are equal if and only if a=c and b=d, that is, the real and imaginary components are equal. Common Core Math Worksheets - K-12. For which values of c does the polynomial have two complex conjugate roots? Answer. Find the square root of a complex number. Simplify the calculation of powers of complex numbers. First, we'll look at the multiplication and division rules for complex numbers in polar form. Express a complex number in polar form whose. Most of these begin with the letters “IM”. Finding the Roots of a Complex Number We can use DeMoivre's Theorem to calculate complex number roots. where the rational function on the right-hand side of the equation satisfies condition 2. Operations with Complex Numbers Some equations have no real solutions. Complex mappings. Convert a complex number from polar to rectangular form. Here r = jzj. You can drag either U or V to see the result of adding other complex numbers. Here I give the formula to find the n-th root of a complex number and use it to find the square roots of a number. Simplify rational expressions with complex numbers in the denominator by using complex conjugates. As another example, consider 35. Notice that any real number is also a complex number, for example 5 = 5 + 0i. We could use the nth root in a question like this:. To practice operations on complex numbers, students complete Complex Number Maze Activity. Complex Numbers Foldable Posted on December 16, 2015 by I Speak Math Complex numbers are amazingly fun to teach in Algebra 2 because it is the first time students have ever seen them. C is the set of all complex numbers, which includes all real numbers. 1 Notation and Terminology In the expression p x the p is called the radical sign. The conjugate of the complex number a + bi is a - bi. Plane Isometries As Complex Functions; Remarks on the History of Complex Numbers. Finding roots of complex numbers. com RR13: Graphs of Square Root Functions - MathOps Graphs of Square Root Functions Quiz & Worksheet - Integer Powers of Complex Numbers | Study. We will start by introducing the complex plane, along with the algebra and geometry of complex numbers, and then we will make our way via differentiation, integration, complex dynamics, power series representation and Laurent series into territories at the edge of what is. fun math practice! math problems on - Powers and roots of complex number, and hundreds of other exercises. 3 Imaginary Numbers Worksheet Templates are collected for any of your needs. 20 Exponents Worksheets. of each other. conjugates. The complex numbers may be represented as points in the plane (sometimes called the Argand diagram). Calculate the value of , and represent the cubic root affixes. This lesson will go over the definitions of powers and roots of numbers. Practice Operations on Complex Numbers. where the rational function on the right-hand side of the equation satisfies condition 2. A complex number has two square roots, three cube roots, four fourth roots, etc. If I said simplify this out you would just combine like terms. Thanks to TES member L Rees-Hughes for uploading the plenary activity. How many times a particular number is a zero for a given polynomial. This is any number that can not be express as an integer divided by an integer, (not counting, no pun intended, imaginary or complex numbers). Complex linear root method roots, how to convert decimal to frACTION ON A TI-86 CALCULATOR, program solving a system in three variables, find the sum of integers between 1 and 100 that are devisible, how do you calculate absolute values on the TI-84 plus graphing calculator, roots of real numbers calculator. Learning how to factor polynomials does not have to be difficult. The resultant complex number is therefore () 1 2 rreiθ 1+θ 2. 35) Theorem. Here a = 1 , b = 10 and c = 169. Complex numbers make 2D analytic geometry significantly simpler. Determine if 2i is a complex number. Now you've seen how imaginaries work; it's time to move on to complex numbers. They determine the missing number in a sequence. The Cubic Formula (Solve Any 3rd Degree Polynomial Equation) I'm putting this on the web because some students might find it interesting. Complex Numbers: an Interactive Gizmo. 5 COMPLEX NUMBERS Learning Targets: 1. We hope you enjoy your adventures with complex numbers and they give you a taste for the exciting mathematics you can discover by choosing Further Mathematics at A-level. Exercise 6. This means that each term only appears once in the denominator, and the root of each term in the denominator is a distinct real number. Express imaginary numbers as bi and complex numbers as [latex]a+bi[/latex] You really need only one new number to start working with the square roots of negative numbers. k Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 2 Name_____ Operations with Complex Numbers Date_____ Period____. The questions could be cut out and students asked to order them or even think of a number to a power that is equivalent. 3i· 4i = 12i 2 = 12(−1) = −12. Express a complex number in polar form whose. The defining property of i. This is not typically the way you want to proceed if you are working by hand, but may be easier for computer solutions (where complex. Powers and roots. Thus in the example just given the roots r 1 = w and r 5 = w 5 are primitive roots of 1. Complex Numbers Name_____ MULTIPLE CHOICE. 7 Polar Form of Complex Numbers In this section, we return to our study of complex numbers which were rst introduced in Section 3. The field of complex numbers. Leave answers in polar form. Irrational and Imaginary Root Theorems Descartes' Rule of Signs More on factors, zeros, and dividing The Rational Root Theorem Polynomial equations Basic shape of graphs of polynomials Graphing polynomial functions The Binomial Theorem. So b 2−4ac=(−3−i) −4×1×(2+i)=9−1+6i−8−4i=2i. Once you find your worksheet, click on pop-out icon or print icon. Complex roots of a. Calculate the value of , and represent the cubic root affixes. Free practice questions for Precalculus - Powers and Roots of Complex Numbers. Excel's powerful mathematical toolkit includes functions for square roots, cube roots, and even nth roots. solve such equations, i. The discovery of analytic geometry dates back to the 17th century, when René Descartes came up with the genial idea of assigning coordinates to points in the plane. The plane representing complex numbers as points is called complex. Imaginary numbers have been a bee in my bonnet for years — the lack of an intuitive. Essential Question: LESSON 2 - COMPLEX NUMBERS. Complex numbers are introduced in algebra courses and the letter is used to denote , and it is called the imaginary unit. Worksheets are Operations with complex numbers, Complex numbers and powers of i, Adding and subtracting complex numbers, Appendix e complex numbers e1 e complex numbers, Properties of complex numbers, Complex numbers, Irrational and imaginary root theorems, Rationalizing imaginary denominators. Geometry and topology of the complex plane. $$ This actually has three solutions, and we can find them using de Moivre's Theorem. Arithmetical root. Some equations may have repeated roots. A given quadratic equation ax 2 + bx + c = 0 in which b 2 -4 ac < 0 has two complex roots: x = ,. The expression b beneath the radical symbol is called the radicand and the small number in the "vee" of the radical symbol is called the index. Problems involving powers of complex numbers can be solved using binomial expansion, but applying De Moivre's theorem is usually more direct. Apply the Pythagorean Theorem to find the hypotenuse of a right triangle. It is easier to write 23 than 2 2 2. Academic Vocabulary Development: Imaginary number-numbers involving the imaginary unit "i" which is defined to be the square root of -1 Real numbers-any number that is a positive number, a negative number or zero. ©z gKyu5ttas pSSo1fNtNwtaRrMeN wL5L5CW. Three part lesson on finding squares, cubes and square roots. How to find the Powers and Roots of Complex Numbers? Using DeMoivre's Theorem to Raise a Complex Number to a Power Raising complex numbers, written in polar (trigonometric) form, to positive integer exponents using DeMoivre's Theorem. S Worksheet by Kuta Software LLC Algebra 2 - Complex Numbers Unit Name_____ Practice Test Period____ Simplify the imaginary numbers. 25 25 1 Factor out 1. See , , and. 2) Solve for ALL solutions (in rectangular form): x3 8 3) Determine the seventh roots of i. 1 Introduction The complex number system is an extension of the real number system. Some examples are shown in Figure 1. In order to use DeMoivre's Theorem to find complex number roots we should have an understanding of the trigonometric form of complex numbers. Powers of 10 Percent Calculations Add and Subtract with Missing Numbers The Math Worksheet Site is provided by Scott Bryce. Complex numbers is vital in high school math. The square root of a number, n, written. (over the complex number system. Football and Other Integer Word Problems. In the case of quadratic polynomials , the roots are complex when the discriminant is negative. The complex numbers are in the form of a real number plus multiples of i. Determine if 2i is a complex number. Complex numbers can be multiplied and divided. a is the real part; b or bi is the imaginary part. Products, Quotients, Powers, and nth Roots. Write complex numbers in polar form. Class Notes. The complex number i is purely algebraic. 8 Modeling with Quadratic Functions. However, through the diligent studies of open-minded mathematicians. Decimal - Easy. Calculate the value of , and represent the cubic root affixes. Create diagrams of simple and complex sentences with these worksheets. 7 Nth roots. : Pianos Sale Dfw Preowned Pianos Pianos Sale Dfw Preowned Pianos. The expression under the radical sign is called the radicand. But, if our numbers are complex that makes finding its power a little more challenging. The number (n. Roots of Complex Numbers Worksheet 1) Determine the fifth roots of 32. Thus, the equation has two roots. All \(n\) of the \(n\)th roots of a complex number \(z\) are evenly spaced around a circle centered at \(0\) and having a radius. So let's try that with a few more problems, just to make it clear that you can do really, really arbitrarily crazy things. Standard Form for a Complex Number a+bi, where a and b are real numbers (a,b∈R). In many cases, these methods for calculating complex number roots can be useful, but for higher powers we should know the general four-step guide for calculating complex number roots. If discriminant is greater than 0, the roots are real and different. com RR13: Graphs of Square Root Functions - MathOps Graphs of Square Root Functions Quiz & Worksheet - Integer Powers of Complex Numbers | Study. Powers and roots. Complex Numbers The square root of negative one, which we now call the imaginary unit, was once thought to be an absurd notion. Roots Of Complex Numbers. Before you start, it helps to be familiar with the following topics: Representing complex numbers on the complex plane (aka the Argand plane). Plassmann's Virtual Classroom. Powers and Roots GCSE revision worksheet? Create your own on Calculators Calculus Christmas Circle Theorems Complex Numbers Conference Congruence. Step 3: Simplify the powers of i, specifically remember that i 2. It states that, for a positive integer n, z n is found by raising the modulus to the n th power and multiplying the argument by n. , to find square roots of negative numbers, we extend the system of real numbers to a new system of numbers known as complex numbers. M y hAnlWlB Yr1itg0ht8sG wr3ezs7eNrvvIexdb. 5 we indicated that it is possible to make sense out of expressions such as or without appealing to a number system beyond the framework of complex numbers. multiplicationanddivision. Solve the following root, expressing the results in polar form. one-step equations with rational numbers; worksheets for variables in expressions; Multiplication and division of Monomial(Rational Expression) solve binomial; adding and subtracting positive and negative numbers worksheet; highest common factor worksheet; India formula for quadratic expression; permutation combination equation. 5th grade word problems worksheets PDF. This lesson will go over the definitions of powers and roots of numbers. Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1. Writing Numbers in Scientific Notation Worksheets These Algebra 1 - Exponents Worksheet are great for teaching students to read and write numbers in scientific notation. With 0 and 1 there exists the constructs with which the number line and all the abstractions of mathematics are possible. Entering text in a document is easy. 2 The residue calculus Say that f(z) has an isolated singularity at z0. I think having a vector (or point ;-) multiplication that behaves like complex multiplication is an interesting idea as it would allow to visualize the basic complex arithmetic operations using points or vectors in an easy way. Imaginary Numbers. Solution of a quadratic equation. Bombelli outlined the arithmetic behind these complex numbers so that these real roots could be obtained. Worksheet 38 (7. Finding the Roots of a Complex Number - Concept. 1 Complex Numbers 1. Do you notice any relationship between the polar forms of two complex numbers and the polar form of their product? Polar Form Polar Form Polar Form Polar Form. Solving Quadratic Inequalities. 1 Roots and Radicals 7. To see the text of a State Standard, hover your pointer over the Standard. Properties of Complex Numbers: Class 2. DEFINITION 5. Frequently there is a number above the radical, like this: n p x. 1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of 2×2 matrices. Find the square root of a complex number. Explore a few powers of complex numbers numerically and decide if there is a pattern Use parametric equations to represent a complex number graphically Using De Moivre's theorem, represent the roots of the complex number graphically. Irrational and Imaginary Root Theorems Descartes' Rule of Signs More on factors, zeros, and dividing The Rational Root Theorem Polynomial equations Basic shape of graphs of polynomials Graphing polynomial functions The Binomial Theorem. Powers And Roots. Complex numbers were invented to enhance the set of real numbers and make it possible that every quadratic equation has a root. However, there is still one basic procedure that is missing from the algebra of complex numbers. The polar form is where a complex number is denoted by the length (otherwise known as the magnitude, absolute value, or modulus) and the angle of its vector (usually denoted by an angle symbol that looks like this: ∠). 16 3 1 Factor the perfect square. Imaginary numbers have been a bee in my bonnet for years — the lack of an intuitive. Such equations often occur in the calculation of compound interest: MathsCentre: Negative and fractional powers: Worksheet: A power, or an index, is used when we want to multiply a number by itself several times. Complex analysis. Answers to Dividing Complex Numbers 1) i 2) i 2 3) 2i 4) − 7i 4 5) 1 8 − i 2 6) 1 10 − i 2 7) − 1 7 + 9i 7 8) 3 2 + 3i 2 9) − 1 5 + i 15 10) − 3 13 + 2i 13 11) 2 5 + 3i 10 12) 4 5 − 2i 5 13) − 27 113 − 47i 113 14) − 59 53 + 32i 53 15) 3 29 + 22i 29 16) − 17 25 − 4i 25 17) 57 89 − 69i 89 18) 41 145 − 28i 145 19) 36. Solving Polynomial Functions Worksheet (pdf) Rational Root Theorem and Finding Zeros Worksheet (doc) Rational Root Theorem and Finding Zeros Worksheet (pdf) Applying Polynomial Functions Worksheet (doc) Applying Polynomial Functions Worksheet (pdf) Modeling with Polynomials Worksheet (doc) Modeling with Polynomials Worksheet (pdf). Exercise 10 B.