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  • Polynomial FunctionsExploring Polynomial FunctionsExamplesModeling Data with Polynomial FunctionsExamples

  • x03569111214y4231262117151922

  • Polynomials and Linear FactorsStandard FormExampleFactored FormExamplesFactors and ZerosExamples

  • Writing a polynomial in standard formYou must multiply:(x + 1)(x+2)(x+3)X3 + 6x2 + 11x + 6

  • 2x3 + 10x2 + 12x2x(x2 + 5x +6)

  • Factor TheoremThe expression x-a is a linear factor of a polynomial if and only if thevalue a is a zero of the related polynomial function.

  • Factors and Zeros-3-2-10123(x (-3)) or (x + 3)(x (-2)) or (x + 2)(x (-1)) or (x + 1)(x 0) or x(x 1)(x 2)(x 3)ZEROSFACTORS

  • Dividing PolynomialsLong DivisionSynthetic Division

  • Long DivisionThe purpose of this type of division is to use one factor to find another.440Just as 4 finds the 10)x - 1x3 + 6x2 -6x - 1The (x-1) finds the (x2 + 7x + 1)

  • Synthetic DivisionWhen dividing by x a, use synthetic division.The Remainder Theorem

  • The Remainder TheoremWhen using Synthetic Division, the remainder is the value of f(a).This method is as good as PLUGGING IN, but may be faster.

  • Solving Polynomial EquationsSolving by GraphingSolving by Factoring

  • Solving by GraphingSet equation equal to 0, then substitute y for 0. Look at the x-intercepts. (Zeros)Let the left side be y1and let the right side be y2. (Very much like solving a system of equations by graphing). Look at the points of intersection.

  • Solving by FactoringSum of two cubes

    (a3 + b3) = (a + b)(a2 ab + b2)

    Difference of two cubes

    (a3 b3) = (a b)(a2 + ab + b2)

  • More on FactoringIf a polynomial can be factored into linear or quadratic factors, then it can be solved using techniques learned from earlier chapters.Solving a polynomial of degrees higher than 2 can be achieved by factoring.

  • Theorems about RootsRational Root TheoremIrrational Root TheoremImaginary Root Theorem

  • Rational Root TheoremWhat are Rational Roots?Ps and Qs . ;)Using the calculator to speed up the process.

  • And the Rational Roots are..P includes all of the factors of the constant.Q includes all of the factorsof the leading coefficient.f(x) = x3 13x - 12p = 12q = 1The possible rational roots are:

  • Test the Possible RootsIn this case all roots are real and rational, but you need onlyto find one rational root. This will become clear later.

  • Since -1, -3, and 4 are the Roots,

    (x + 1), (x + 3), and (x 4)

    are the factors.Multiply to show that

    (x+1)(x+3)(x-4) = x3 13x 12 (x+1)(x2 x 12) x3 x2 12x +x2 x 12 x3 13x 12

  • Irrational Root TheoremThese are called CONJUGATES.

  • Imaginary Root TheoremThese are called CONJUGATES.

  • The Fundamental Theorem of AlgebraIf P(x) is a polynomial of degree with complex coefficients, then P(x) = 0 has at least one complex root.

    A polynomial equation with degree n will have exactly n roots; the related polynomial function will have exactly n zeros.

  • The Binomial TheoremBinomial Expansion and Pascals Triangle

    The Binomial Theorem

  • PASCALS TRIANGLE11 11 2 1 1 3 3 11 4 6 4 1 1 5 10 10 5 11 6 15 20 15 6 11 7 21 35 35 21 7 11 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 1

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