Learn how to write a polynomial function with rational coefficients when given irrational zeros! Mario's Math Tutoring explains the Irrational Conjugates Theorem, which states that if a polynomial has rational coefficients and an irrational zero (like 3 + √2), then its conjugate (3 - √2) must also be a zero. We'll go through two detailed examples, demonstrating how to convert zeros into factors, multiply these factors, and use the sum and difference pattern to simplify expressions, ensuring your final polynomial has only rational coefficients. In this video, you'll discover: 0:00 Introduction to Writing Polynomial Functions from Zeros 0:20 Irrational Conjugates Theorem Explained 0:50 Example 1: Zeros are -4 and 3 + √2 1:00 Identifying the Conjugate Pair (3 - √2) 1:30 Converting Zeros to Factors (x - zero) 2:00 Distributing Negatives into Parentheses 2:30 Grouping Terms to Form a Sum and Difference Pattern 3:00 Applying the Difference of Two Squares Concept 3:40 Squaring a Binomial and Simplifying 4:30 Multiplying the Remaining Factors 5:00 Combining Like Terms to get the Final Polynomial (f(x) = x³ - 2x² - 17x + 28) 5:50 Example 2: Zeros are 7 and -2 + √3 6:00 Identifying the Conjugate Pair (-2 - √3) 6:30 Converting Zeros to Factors 6:50 Distributing Negatives into Parentheses 7:20 Grouping Terms for Sum and Difference Pattern 7:50 Applying the Difference of Two Squares and Squaring a Binomial 8:40 Simplifying and Combining Like Terms 9:00 Multiplying the Remaining Factors 9:40 Combining Like Terms to get the Final Polynomial (f(x) = x³ - 3x² - 27x - 7) Master the process of constructing polynomial functions from their irrational zeros! #PolynomialFunctions #RationalCoefficients #IrrationalZeros #ConjugatePairs #Algebra2 #Precalculus #MathTutorial #MathHelp * Organized List of My Video Lessons to Help You Raise Your Scores & Pass Your Class. Videos Arranged by Math Subject as well as by Chapter/Topic. (Bookmark the Link Below) ➡️JOIN the channel as a CHANNEL MEMBER at the "ADDITIONAL VIDEOS" level to get access to my math video courses(Algebra 1, Algebra 2/College Algebra, Geometry, and PreCalculus), midterm & final exam reviews, ACT and SAT prep videos and more! (Over 390+ videos)