Solve polynomial equations

Learning Objectives Apply the Rational Root Theorem to identify all potential rational roots of a polynomial equation. Use synthetic division to test potential roots and depress a polynomial to a lower degree. Factor polynomials completely using a combination of techniques, including grouping and sum/difference of cubes. Solve polynomial equations of degree three and higher to find all real and complex roots. Apply the Fundamental Theorem of Algebra to determine the total number of roots a polynomial equation must have. Use the Conjugate Root Theorem to find pairs of complex roots. Ever wondered how engineers design the perfect curve for a roller coaster or how economists model profit to find break-even points? 🎢 It all starts with solving polynomial equations! This tutori...

TermDefinitionExample Polynomial EquationAn equation in which a polynomial expression is set equal to zero. The general form is a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 = 0.x^3 - 6x^2 + 11x - 6 = 0 is a polynomial equation of degree 3. Root (or Zero)A value of the variable that is a solution to the polynomial equation. Graphically, the real roots are the x-intercepts of the polynomial function.For the equation x^2 - 9 = 0, the roots are x = 3 and x = -3. DegreeThe highest exponent of the variable in a polynomial. The degree determines the total number of roots (including complex and repeated roots) the equation has.The polynomial P(x) = 5x^4 - 2x^3 + 8 has a degree of 4. Synthetic DivisionA shorthand method for dividing a polynomial by a linear factor of the form (x - c). It is used...

The Rational Root Theorem For a polynomial P(x) = a_n x^n + ... + a_0 with integer coefficients, any rational root must be of the form \frac{p}{q}, where p is a factor of the constant term a_0 and q is a factor of the leading coefficient a_n. Use this theorem to create a finite list of all possible rational roots. This narrows down the search for the first root, which can then be tested with synthetic division. The Fundamental Theorem of Algebra A polynomial equation of degree n, where n > 0, has exactly n roots in the complex number system. These roots may be real, complex, and/or repeated. This theorem tells you how many solutions to look for. If you have a degree 4 polynomial, you must find 4 roots to be finished. The Conjugate Root Theorem If a polynomial P(x) h...