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4.3 – Location Zeros of Polynomials

4.3 – Location Zeros of Polynomials. At times, finding zeros for certain polynomials may be difficult There are a few rules/properties we can use to help us at least determine some rough information pertaining to the zeros. Rational Zero Theorem.

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4.3 – Location Zeros of Polynomials

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  1. 4.3 – Location Zeros of Polynomials

  2. At times, finding zeros for certain polynomials may be difficult • There are a few rules/properties we can use to help us at least determine some rough information pertaining to the zeros

  3. Rational Zero Theorem • For a polynomial f(x) = anxn + an-1xn-1 + … + a1x + a0, then any rational zero must be of the form (p/q), where p is a factor of the constant term (a0) and q is a factor of the leading coefficient an • All zeros are ratios of the constant and leading coefficient

  4. Example. For the function f(x) = 2x3 + 5x2 – 4x -3, list the potential rational zeros.

  5. Descartes’ Rule of Signs = for a polynomial, a variation in sign is a change in the sign of one coefficient to the next • 1) The number of positive real zeros is either the number of variation in sign or is less than this number by a positive integer (MAXIMUM, +) • 2) The number of negative real zeros is the number of variations in the sign of f(-x) or less than this by a positive integer (MAXIMUM, -)

  6. Example. For the function f(x) = 2x3 + 5x2 – 4x -3, determine the number of possible positive and negative zeros. • Example. For the function f(x) = x3 – 6x2 + 13x – 20, determine the number of possible positive and negative zeros.

  7. Intermediate Value Theorem = For a polynomial f(x), if f(a) and f(b) are different in signs, and a < b, then there lies at least one zero between a and b • Most useful!

  8. Example. Show that f(x) = x3 + 3x – 7 has a zero between 1 and 2.

  9. Multiplicity of Zeros/Roots • Suppose that c is a zero for the polynomial f(x) • For (x – c)k, or a root of multiplicitity k, the following rules are known: • The graph of f will touch the axis at (c,0) • Cross through the x-axis if k is odd • Stay on the same side of the x-axis if k is even • As k gets larger than 2, the graph will flatten out

  10. Example. Sketch the graph of: • f(x) = (x + 2)(x + 1)2(x – 3)3

  11. Conjugate Pairs • Recall, if you have the complex number a + bi, then the conjugate is a – bi • If a polynomial f(x) has the imaginary zero a + bi, then the polynomial also has the conjugate zero, a – bi • If x – (a + bi) is a factor, so is x – (a – bi)

  12. Example. Construct a fourth degree polynomial function with zeros of 2, -5, and 1 + i, such that f(1) = 12.

  13. Conclusion: Still best to use, when applicable, any graphing utility to find the zeros of functions. • However, when you’re stuck, these properties give us a wide range of potential values to choose from, and methods to narrow down the list of “candidate zeros.”

  14. Assignment • Pg. 337 • For the problems, do the following: • 1) List the potential zeros • 2) Show # of + or – zeros • 3) Use your graphing calculator to identify the zeros • #58, 60, 64

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