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Real Zeros of a Polynomial Function

Explore solving polynomial equations, applying Descartes’ rule, and finding polynomial equations. Learn to determine positive and negative roots, use Rational Root Theorem, and solve using techniques like synthetic division and quadratic formula. Practice solving functions like 3x^5 + 2x^4 + 15x^3 + 10x^2 – 528x - 352 or 3x^4 + 5x^3 + 25x^2 + 45x - 18. Understand finding complex and irrational roots in conjugate pairs and identifying all roots as solutions of x. Prepare for quizzes and tests with provided assignments.

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Real Zeros of a Polynomial Function

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  1. Real Zeros of a Polynomial Function Objectives: Solve Polynomial Equations. Apply Descartes Rule Find a polynomial Equation given the zeros.

  2. 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.

  3. 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

  4. 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

  5. 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

  6. 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)

  7. Find the zeros for3x5 + 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.

  8. 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.

  9. 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

  10. 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.

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