Mr. Serendip's ATC Math Classes - Home



Pre-calculus Chapter 4 Polynomials, Rational and Radical Equations Polynomials: have positive integer exponents – standard form is in order of HIGHEST EXPONENT to lowest, with constant last.Quadratics – second order polynomial (x2) – solve by setting equal to zero FACTORING - find a pair of factors of the constant that add up to the coefficient of the x term – i.e for x2 + 3x – 10, +5 and -2 are factors of 10 that add up to +3, so (x+5) (x-2) are the factors… these yield ROOTS of -5 and 2. A ROOT is when you set a factor = 0 and solve for x. It is an x-intercept if it is a real number, but not if PLETING THE SQUARE – make any quadratic into a Perfect Square Trinomial then solve by taking square root of both sides. First, put in standard form. Make sure lead coefficient (coefficient of x2) is 1, then move constant to the other side of the equation leaving just the x2 term and x term. Then add (b2)2 to both sides (b is coefficient of x term). This will make a perfect square trinomial. Its factors will be: (x + b2)2 if b is positive or (x - b2)2 if b is negative. Take square root of both sides ( ± !!) and solve for x.Quadratic Formula - x=-b±b2-4ac2a end of story! Always works, just keep track of your signs! Remember to simplify under the radical if you can.Higher order Polynomials:To solve, you must use a combination of root theorems to get to final answers. Fundamental Theorem of Algebra – tells you TOTAL number of roots = highest exponent (but this includes imaginaries, multiplicities, etc.)Rational Roots Theorem – tells you any actual rational roots will be in the form of pq where p is factors of the constant term, and q is factors of the lead coefficient (coeff. Of highest exponent of x). There may be many possible, and few or no actual! Check them by synthetic division.Remainder Theorem – tells you if you do synthetic division on a polynomial and get a zero remainder, the thing you divided by is a root! (so (x-that) is a factor). AND whatever you have left is the other factor, and can then be factored or examined by other means for further roots. AND if you DO get a remainder, the remainder is the value of the f(the divisor).Descartes’ Rule of Signs – tells you the number of possible positive real roots = number of times sign changes in function over consecutive integers (like x2 – 3x + 2 is two sign changes, so 2 positive roots) or less than that by an even number (so could be zero positive roots), and number of negative real roots is the same but for f(-x).Conjugate Root Theorem – tells you that if you find any irrational or imaginary root, its opposite MUST also be a root.By applying all of these in an orderly sequence you can solve all the roots of any polynomial.23262345171846In such a rare case, you are probably stuck – use a calculator, computer, or calculusIn such a rare case, you are probably stuck – use a calculator, computer, or calculus1616659547908527724611185062Conjugate root theorem!00Conjugate root theorem!27212544023360Check Descartes’ Rule of SignsOnly try possible options00Check Descartes’ Rule of SignsOnly try possible options-3364995252314Means the polynomial is the product of unfactorable quadratics0Means the polynomial is the product of unfactorable quadratics3218692018995-2487171667866Factor our any GCF?Factor our any GCF?2999231499616010140952316810yes00yes-4165602229790Is it quadratic?0Is it quadratic?02767025no00no-4096514330597No rational roots0No rational roots10241284103827Not quadratic, see if you can get another rational root!Not quadratic, see if you can get another rational root!21882102748610yesyes21907503405200Is what’s left over quadratic?0Is what’s left over quadratic?13455653262935Got a rat. root0Got a rat. root-3950213160165Look for rational rootspq test by synthetic div.0Look for rational rootspq test by synthetic div.37426901862150DONE!DONE!286755820775172 roots002 roots16819881850009Solve by factoring, quadratic formula etc.Solve by factoring, quadratic formula etc.1675181512064This # is your goal!00This # is your goal!1163117950976-321869651053Fundamental Theorem Total # of roots = ?0Fundamental Theorem Total # of roots = ?Solving ALL roots of a polynomial will often be a recursive process:Of course, you can easily go backwards from roots to factors (if 3 is a root, then (x-3) is a factor; if 3 is a root then (x-3) is a factor – and from factors you can go backwards to solve a polynomial -- if factors are (x + 2i) (x – 2i) (x + 3) then multiply all terms by all terms… (x + 2i)(x – 2i) = x2 + 4 (x2 +4)(x +3) = x3 + 3x2 +4x +12 Rational Equations: where variable is in denominator of a fraction – solved by using Least Common Denominators to simplify. Look for opportunities to simplify early – do numerators and denominators share common factors that can be cancelled out? Would a common denominator be simpler if you moved terms in equation around first? When its as simple as you can get it, multiply the denominators together – that’s your common denominator. Then multiply every term on both sides by that. This may be easier if you leave the common denominator expressed as a product of factors [like using (n+1)(n-1) instead of (n2 -1) ]Example: 9b+5= 3b-3 common denominator is (b+5)(b-3) multiply both sides and get9b-27 = 3b+15 simplify to 6b = 42 b=7 REMEMBER TO CHECK that solutions do not yield a zero denominator in original! If they do, discount them!Radical Equations: have variable under a radical sign. If square root, then square both sides and solve for x. If both sides have a square root, then you may have to F.O.I.L. one side!REMEMBER TO CHECK that solutions are not extraneous! Plug them into the original – they don’t always work.Ex. 3x+10= x+11-1 square it to get 3x +10 = (x+11-1)( x+11-1)Which simplifies to 3x + 10 = x + 11 - 2x+11 + 1 this becomes: 2x – 2 = -2x+11Then divide by -2: 2x-2-2 = x+11 square both sides x2 - 2x + 1 = x + 11 So x2 – 3x – 10 = 0 this factors to (x+2)(x-5) so x = -2 or 5 but if you plug in 5 it doesn’t work, so the answer is x = -2 ................
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