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Polynomial FunctionsExploring Polynomial FunctionsExamplesModeling Data with Polynomial FunctionsExamples
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