Write a polynomial from its roots

Learning Objectives Write a polynomial function of least degree given its real, rational roots. Construct a polynomial when given roots with specified multiplicities. Apply the Complex Conjugate Root Theorem to find all roots and write the corresponding polynomial with real coefficients. Utilize the Irrational Conjugate Root Theorem to construct a polynomial with rational coefficients from its irrational roots. Determine the specific leading coefficient of a polynomial using a given point on its graph. Combine these skills to write a polynomial from a mix of real, irrational, and complex roots. Ever solved a puzzle by working backward? 🧩 This lesson teaches you how to reverse-engineer a polynomial function just by knowing where it crosses the x-axis! We will explore the fu...

TermDefinitionExample Root (or Zero)A value 'c' for the variable x such that the polynomial P(x) evaluates to zero, i.e., P(c) = 0. Graphically, real roots are the x-intercepts of the function.For the polynomial P(x) = x² - 4, the roots are x = 2 and x = -2 because P(2) = 0 and P(-2) = 0. Factor TheoremA cornerstone theorem stating that if 'c' is a root of a polynomial P(x), then (x - c) is a factor of P(x). Conversely, if (x - c) is a factor, then 'c' is a root.Since x = 5 is a root of P(x) = x - 5, the Factor Theorem tells us that (x - 5) is a factor. MultiplicityThe number of times a particular root corresponds to a factor in a polynomial. A root with multiplicity 'k' means the factor (x - c) appears 'k' times.In P(x) = (x - 3)²(x + 1),...

Linear Factorization Theorem P(x) = a_n(x - c_1)(x - c_2)...(x - c_n) This theorem states that any polynomial P(x) of degree n > 0 can be expressed as a product of its n linear factors. 'a_n' is the leading coefficient and c₁, c₂, ..., cₙ are the complex roots of the polynomial. Product of Complex Conjugate Factors (x - (a + bi))(x - (a - bi)) = x^2 - 2ax + (a^2 + b^2) Use this formula to quickly multiply the factors corresponding to a complex conjugate pair of roots. This is essential for ensuring the resulting polynomial has real coefficients. Product of Irrational Conjugate Factors (x - (a + \sqrt{b}))(x - (a - \sqrt{b})) = x^2 - 2ax + (a^2 - b) Use this formula to efficiently multiply factors from an irrational conjugate pair of roots. This ensures t...