Transcript Slide 1
Real Zeros of a Polynomial Function Objectives: Solve Polynomial Equations. Apply Descartes Rule Find a polynomial Equation given the zeros. Solving Polynomial Functions • Fundamental Theorem of Algebra: All polynomial functions of degree ‘n’ will have ‘n’ roots. • Descartes Rule Determines the possible number of positive and negative roots by looking at sign changes in the function Count sign changes in the original function: tells number of maximum positive real roots. Substitute a negative x in for each x, simplify, then count sign changes: tells number of maximum number of negative real roots. Determine the number of roots and the possible number of positive and negative roots. • F(x) = 2x3 + x2 + 2x + 1 3 roots No sign changes: no positive real roots Substitute a negative x in for x and count sign changes 2(-x)3 + (-x)2 + 2(-x) + 1 -2x3 + x2 – 2x + 1: 3 sign changes 3 or 1 negative real root F(x) = 6x4 – x2 + 2 • • • • 4 roots 2 or 0 positive real roots: 2 sign changes 6(-x)4 – (-x)2 + 2 6x4 – x2 + 2: 2 or 0 negative real roots: 2 sign changes in translated function Rational Root Theorem • All rational roots will come from Factors of the last term / factors of the first term List the potential rational zeros of the polynomial function. F(x) = 3x5 – x2 + 2x + 18 Factors of the last term: +- 1,2,3,6,9,18 Factors of first term: 1,3 Possible rational roots: 1, 2, 3, 6, 9, 18, 1/3, 2/3, -1, -2, -3, -6, -9, -18, -1/3, -2/3 Solving Polynomial Equations • 1. Set equal to zero • 2. Use POLY or Graphing to find rational zeros. • 3. Use synthetic division to get equation to quadratic form. • 4. Use quadratic formula to solve for irrational and complex roots • (complex and irrational roots will always occur in conjugate pairs) Find the zeros for 3x5 + 2x4 + 15x3 + 10x2 – 528x - 352 • • • • • 5 answers: at least one positive root 4, 2, or 0 negative roots Use poly to find “nice” answers 4i, -4i, -2/3 Use synthetic division to get to a quadratic expression then solve for the two remaining solutions using quadratic formula. F(x) = 3x4 + 5x3 + 25x2 + 45x - 18 • Use POLY • -3i, 3i, -2, 1/3 • Do not need quadratic formula since all answers came out nice. Find a function with zeros of 2, 1+i, 2i • Write all zeros as solutions of x. x = 2, x = 1 + i, x = 1 – i, x = 2i, x = -2i • Write each in factor form x – 2, x – 1 – i, x – 1 + i, x – 2i, x + 2i • Multiply factors together (multiply conjugates first) (x-2)(x-1-i)(x-1+i)(x-2i)(x+2i) • Product is function with given zeros Assignment • Page 374 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 115 • Page 382 7, 11, 13, 17, 21, 25, 29, 33, 37, 39 Quizzes and test sometime before next Monday.