2'4 Finding Rational Zeros of Polynomials

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2.4 Finding Rational Zeros of Polynomials

• Some polynomials can be broken down into the
  product of factors
• to determine where f(x) 0.
• f(x) x3 x2 2x
• x( x2 x 2)
• x(x-2)(x1) 0 when x 0, 2, or -1
• f(x) x3 2x2 x 2
• x2(x2) 1(x2)
• (x2)(x2-1)
• (x2)(x1)(x-1) 0 when x -2, -1, and 1
• Common Factoring Formulas x3 y3 (x-y)(x2
  xy y2)
• x3 y3 (xy)(x2 xy y2)
• If given enough information, the zeros can be
  used to determine

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• If a polynomial is not easily factored, long and
  synthetic division can be used
• to find its zeros
• Long Division
• x 2 6x3 19x2 16x 4
• Remainder of zero implies that x 2 is a factor
  of the polynomial, resulting in
• the following factorization
• 6x3 19x2 16x 4 (x-2)(6x2 7x 2)
• (x-2)(2x-1)(3x-2)
• x 1 x2 3x 5

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• The remainder of zero implies that x 2 is a
  factor or the polynomial, resulting in the
• following factorization
• 2x4 7x3 4x2 27x 18 (x-2)(2x3 11x2
  18x 9)
• Continuing the factorization, 2x3 11x2 18x
  9 could be divided by x 3
• -3 2 11 18 9
• -6 -15 -9
• 2 5 3 0
• Since the remainder is zero, x 3 is another
  factor. This simplifies the polynomial to
• (x-2)(2x3 11x2 18x 9) (x-2)(x3)(2x2 5x
  3)
• (x-2)(x3)(2x3)(x1)