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Unit 3.3Polynomial Equations Continued Objectives Divide polynomials with synthetic division Combine graphical and algebraic methods to solve polynomial equations Use the Fundamental Theorem of Algebra to find the number of complex solutions of a polynomial equation Find the complex solutions of polynomial equations Division of Polynomials; Synthetic Division Suppose f(x) is a cubic function and we know that a is a solution of f (x) = 0. We can write (x - a) as a factor of f (x), and if we divide this factor into the cubic function, the quotient will be a quadratic factor of f (x). If there are additional real solutions to f(x) = 0, this quadratic factor can be used to find the remaining solutions. 4 3 Divide 𝑥 + 6𝑥 − 5𝑥 + 4 by x + 2 Long Division Synthetic Division Example If x = 2 is a solution of x3 + 6x2 – x – 30 = 0, find the remaining solutions. Solution Example (cont) Thus the quotient is ________________, with remainder 0, so x – 2 is a factor and Y 18 12 6 X -7 -6 -5 -4 -3 -2 -1 0 -6 -12 -18 The solutions are x = ___, __, and ___. -24 -30 -36 1 2 3 Multiplicity A graph touches but does not cross the x-axis at a zero of even multiplicity, and it crosses the xaxis at a zero of odd multiplicity. Example The weekly profit for a product is P(x) = – 0.1x3 + 11x2 – 80x – 2000 thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even, graph the function using a window representing up to 50 thousand units and find one x-intercept of the graph. We use the viewing window [0, 50] by [–2500, 8000]. Example (cont) b. Use synthetic division to find a quadratic factor of P(x). Solution Example (cont) c. Find all of the zeros of P(x). Solution Example: Determine all possible rational solutions: 3𝑥 3 + 3𝑥 2 − 9𝑥 + 12 = 0 Example Solve the equation 2x4 + 10x3 + 13x2 – x – 6 = 0.