Humboldt State University
Mathematics Department Colloquium *
click on * to go to next section
September 24, 2017
Solving Polynomial Equations:
Visualization from Linear to Cubic,
from Rational to Complex Numbers
Martin Flashman
Professor Emeritus of Mathematics
Humboldt State University
Arcata, CA 95521
flashman@humboldt.edu
flashman.neocities.org
Abstract:
From Euclid to Descartes, solving equations and visualization of
relations between numbers have been a central theme of mathematics. In
school mathematics these problems start formally in middle school and
continue to the solution of the general quadratic equation where complex
numbers are usually introduced. Cubic equations are more challenging
and only special examples are treated until calculus where applications
of the intermediate value theorem are used to explain the solution of
cubic equations. Rarely is any explicit formulation of the general
solution of a cubic polynomial given or the nature of these solutions
explored extensively.
Using some recent work with Geogebra on mapping diagrams, an alternative
to graphs, Professor Flashman will provide visualizations for both the
algebra used to solve linear and quadratic equations and the relation of
quadratic and cubic equations to complex number solutions.
0. Preface:*
- Mapping Diagrams from A(lgebra)
B(asics) to C(alculus) and D(ifferential) E(quation)s. A Reference and Resource Book on Function Visualizations
Using Mapping Diagrams (In Development-Draft Version 8-2017)
- GeoGebra "Books"
Tom Edgar and N. Chris Meyer
The College Mathematics Journal
Vol. 48, No. 2 (March 2017), pp. 90-96
http://dx.doi.org/10.4169/college.math.j.48.2.90
I. What equations? *
Linear, quadratic, and cubic equations.
What does it mean to solve an equation?
Find any and all numbers in the solution domain that make the equation a
true statement when substituted for the equation's variable.
- Linear: $a_1 x+ a_0= 0$
Example: $3x -6 = 0.$
Solution: $x = 3$
Check! OOOPS: $3\cdot 3-6=9-6=3 \ne 0$. Correct solution: $x=2$.
- Quadratic: $a_2x^2 + a_1 x+ a_0= 0$
Example: $x^2 - 5x +6 = 0.$
Solution: $x = 3, x = 2$
Check!
- Cubic:$a_3x^3 +a_2x^2 + a_1 x+ a_0= 0$
Example: $x^3 - 5x^2 +6x = 0.$
Solution: $x = 3, x = 2, x=0$
Check!
II. Solution Domains:*
- Natural Numbers: $N=\{0
(?),1,2,...
\}$
![](data:image/png;base64,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)
Example: $3x -6 = 0.$
Solution: $3x -6 = 0$
$+6 = +6$
$3x= 6$
$\frac 13 \cdot (3x)=\frac 13 \cdot (6)$
$x = 3$
Check!
- Integers:
$Z=\{0,1,-1,2,-2,...\}$ *
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAABZgAAADYCAIAAADprxFYAAAOJElEQVR4nO3dX2Td9//A8TcRx1FHHCpqaqLsKqUXo6amojdVURMVaqqqZkwvejG9mdGYXZSoqqoauahdVIVcTEXN2KoqJsLMVFWNXMxsUaPSqInj8734TPWbpLVff9+c9/vVz+NxtxPlvbze5/PneU7OSRUAAABAECn3AgAAAAD+LSEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMIQMgAAAIAwhAwAAAAgDCEDAAAACEPIAAAAAMry9OnTl/1IyAAAAAAKcv78+dHR0ZWVlS1/KmQAAAAApXj69Ono6GhK6WUtQ8gAAAAACrKysvKKliFkAAAAAGV5RcsQMgAAAIDivKxlCBkAAABAibZsGUIGAAAAUKjNLSP98MMP5wEAAACKdO7cuZGRkbplrK2tpfPnzycAAACAUp05c6bVao2NjVVVle7cuTOVydjYWEppbGws1wKYMoViGEQJTKEEplACUyiBKZTAFEpgCiUwhRI0fArP35Gxd+/etbW1nJ+RUb8ZZGpqKuMaMIVCGEQJTKEEplACUyiBKZTAFEpgCiUwhRI0eQp//vln/RkZe/fu/eczMjKupsmTKIcpFMIgSmAKJTCFEphCCUyhBKZQAlMogSmUoLFT2FwxKiEDUyiEQZTAFEpgCiUwhRKYQglMoQSmUAJTKEEzp7BlxaiEDEyhEAZRAlMogSmUwBRKYAolMIUSmEIJTKEEDZzCyypGlTdk1J8zeufOnYxrwBQKYRAlMIUSmEIJTKEEplACUyiBKZTAFErQtCm8omJUeUMGAAAAwIuePn36iopRCRkAAABAUaampl5WMSohAwAAACjN2tray34kZAAAAABhCBkAAABAGEIGAAAAEIaQAQAAAIQhZAAAAABhFBQyHj9+nF4i99Ka69atW0aQxdWrV/fv399ut9vt9rvvvnvp0qVer5d7UY1z586dY8eOdTqdwcHBdrt96NChGzdu5F5Uo507d87haFt9//33R48erff80NDQ8ePH7927l3tR2PnZOAuUwBVRadwdZOFOebOC/s/n5uaMpyirq6u7d+82gj7r9XpHjhzZ/Cw4ePDgs2fPcq+uQb788sstD0cnT57MvbSGmp+fdzjaVtevX99yz1+5ciX30hrNzs/FWSA7V0QFcneQizvlzQr6P//0009TSpcuXcq9EP5x9uxZz5D+m56eTimNjIzMz8/3er1er3f37t09e/aklD7//PPcq2uKe/fupZRardaFCxceP35cVdWTJ0+mp6cHBwdTSteuXcu9wMaZm5trtVoOR9tnbW2t2+2mlD766KNff/21qqrl5eWTJ0+mlLrdrtc/c7Hzc3EWKIErogK5O8jFnfJmBW3BsbGxlNLS0lLuhVBVVbW0tJRS2rVrl0NVn73zzjsppfv377/44C+//JJS2r17d65VNc2xY8dSSl999dWGx2dmZlJK+/bty7KqZnry5MmZM2dSSu122+Fo+9y8eTOlNDExseHxAwcOpJS++eabLKtqMjs/L2eBErgiKo27g4zcKW9Wyhbs9XqDg4Pdbjf3Qqiqqur1eqOjoyml2dlZh6pC1C8N5V5FU3Q6nZTS+vr6hsd7vZ5B9NnQ0FD9itzi4qLD0fY5ceJESmlubm7D4zdu3EgpffLJJ1lW1WR2fl7OAiUzgizcHWTkTnlLpWzBH3/8MaU0OTl58eLFffv2tVqtTqfz4Ycf/vTTT7mX1kT134WOj49XVeVQVYKFhYWU0oEDB3IvpOnW19dTSp1OJ/dCGqTVan322Wdra2uVw9F2qt95seGVz6qq7t+/n1J67733sqyqyez8MjkLZOeKKBd3Bxm5U95SKVvw6tWraSsDAwOzs7O5V9csjx49arVaO3bs+O233yqHqtyePXs2MzMzPDw8ODh49+7d3MtpuvrV6RMnTuReSIPUf51eczjaPi97/fnvv/9225aFnV8mZ4GMXBFl5O4gL3fKWyplCx4/fjyltHPnztu3b9cfKra6ujozM9Ptdlut1oMHD3IvsEHqP8G6fPly/Z8OVRlNTEzUv/92u72wsJB7OU23urq6Z8+eTqfz8OHD3GtpKIej7VN/hOGWP/Iu7uzs/EI4C2Tkiigvdwd5uVPeUilb8O23304p3b59e8PjdX/6+OOPs6yqgepPsdq/f//zRxyqMtq/f//4+Hj9Ad07d+78+uuvc6+ouXq93uHDh4eHh10/ZeRwtH1e8butX/Pp83p4kZ1fAmeBvFwRZeTuIDt3ylvqxxbc8p0wz736366srKSU3nrrrT6s8832b6awsrLS7XYHBgZe/DNph6r/odd+Lty7d294eDildOvWrb6t9k31elOYnJwcGRn5/fff+7nUN9jrTcHhaPvU3/G55Y+Sd2TkZueXwFmgEK6I+szdQckafqdcesio//ng4GAf1vlm+zdTmJycTJu+mtuh6n/o//Nc+Pbbb9N/53Bez/91Cuvr6xMTEzt27Hj06FH/V/umer3ngsPR9qm/I2PzZ2TUH204NDSUZVXU7Py8nAVK44qon9wdFC41+E659C1Yf81Vu93OvZBGePWthQNWXvVzobGHqlweP358+PDhlNLFixdzrwVXTtvo0KFDKaXNf2f78OHDlNL777+fZVXU7PyMnAUK5Iqon9wdlKzhd8qlbL7689L/+uuvDY/Xl1AHDx7MsqqmcagqQf26aP2h0C/y3QH9t7CwsHv37pTShQsXcq+FqnI7t51Onz6dUpqbm9vw+OzsbPIdDbnZ+bk4C+TliqgE7g5K4E55S6VsvvptS88/C/e5L774IqU0PT2dZVVUrp/67sSJEymlqampDY9fuXIlpfTBBx9kWVUDzc/Pt1qtoaGh+fn53GvhHw5H2+fmzZsppSNHjmx4/MiRIymlGzduZFkVNTs/C2eB7FwRFctBqc/cKW+plC24uLiYUup2u7Ozs/WXyjx58uTy5cvtdntkZOTZs2e5F9hcDlV99vPPP9dvErt+/Xr99+pra2vXrl3rdDoDAwNLS0u5F9gIDx48aLfbw8PDjf1GqzI5HG2f9fX1brebUjp16tTy8nJVVcvLy6dOnUopDQ0Nra6u5l5go9n5/ecsUAJXRMVyUOozd8pbKmgL1nl1g+Hh4Rc/I5f+c6jqv/prrjYYGBjwomjfPP++em+kLIpf/raanp7ecrf7msPs7Pz+cxYohCuiMnkW9J875c3K2oJ37949evRou90eGBgYGRk5e/bsH3/8kXtRTedQlcXi4uLExET9msOuXbtOnz7d5ONU/9VfRekStjR++dvt1q1b4+Pj9ZGn0+mMj49/9913uReFnZ+Bs0A5XBEVyLMgC3fKG9iCAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYQgZAAAAQBhCBgAAABCGkAEAAACEIWQAAAAAYfwHr+Z+Pqi676cAAAAASUVORK5CYII=)
Example: $3x +6 = 0$.
Solution: $3x +6 = 0$
$ -6 = -6$
$3x= -6$
$\frac 13 \cdot (3x)=\frac 13 \cdot (-6)$
$x = -2$
Check!
- Rational numbers: $Q=\{\frac
pq, p,q \in Z, q\ne
0\}$. *
![](data:image/png;base64,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)
Example: $3x - 4 = 0$.
Solution: $3x - 4 = 0$
$4=4$
$3x=4$
$\frac 13 \cdot (3x)=\frac 13 \cdot (4)$
$x = \frac 43$
Check!
- Real Numbers: $R= \{ x: x= \pm ***.***..., (infinite)\ decimal \}$
![](data:image/png;base64,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)
Example: $x^2 - 2 = 0$.
Solution: $x = \sqrt 2, x= -\sqrt 2$
Check!
- Complex Numbers: $C = \{a+bi: a,b \in R, i^2=-1\}$ *
![](data:image/png;base64,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)
Example: $x^2 + 4 = 0$.
Solution: $x = 2\sqrt {-1} = 2i, x= -2\sqrt {-1}=-2i ;\ i^2 = -1$
Check!
III Fundamental Theorem of Algebra *
If $a_nx^n+ ... +a_2x^2 + a_1 x+ a_0= 0$ is an equation with
complex number coefficients then there are exactly n complex number
solutions (allowing for possible repeated roots).
Or
Every non-constant single-variable polynomial with complex coefficients
has at least one complex root. This includes polynomials with real
coefficients, since every real number is a complex number with an
imaginary part equal to zero.
First stated in some form as early as about 1629 (Albert Girard),
first proven (with a "gap") by Carl Friedrich Gauss (1777-1855) in
his Ph.D. Thesis (1799) and a complete proof in its current form
was given by Argand in 1816. [Just over 200 years ago!]
See Wikipedia: https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
IV Solving Equations Algebraically. *
[For clever solutions factoring is a key tool. FACT: If $a \times b = 0$ then either $a=0$ or $b=0$.]
- Linear: $a_1 x+ a_0= 0$
Example: $3x -6 = 0$.
$3(x-2)=0$
....
Solution: $x = 2$
- Quadratic: $a_2x^2 + a_1 x+ a_0= 0$
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAeYAAAHKCAIAAABZn3/VAAAgAElEQVR4nO2dfVxUVeL/bw0wsoKCD4TKKpmILuGoKCIhYLE+ESESpWg0IZKPEdKqGYIaPpQPaCSIIKIikESmRCxajgI+hGxEfV3WL7K4YOKaXzbz688vS9P8/rg1e2Z4us65zrlXPu/X+59mDsMF57wvzT1nhtMBAACQCRzrAwAAACAUJBsAAGQDkg0AALIByQYAANmAZAMAgGxAsgEAQDYg2QAAIBuQbAAAkA1INgAAyAYkGwAAZAOSDQAAsgHJBgAA2YBkAwCAbECyAQBANiDZAAAgG5BsAACQDUg2AADIBiQbAABkA5INAACyAckGAADZgGQDAIBsQLIBAEA2INkAACAbkGwAAJANSDYAAMgGJBsAAGQDkg0AALIByQYAANmAZAMAgGxAsgEAQDYg2QAAIBuQbAAAkA1INgAAyAYkGwAAZAOSDQAAsgHJBgAA2YBkAwCAbECyAQBANiDZAAAgG5BsAACQDUg2AADIBiQbAABkA5INAACyAckGAADZgGQDAIBsQLIBAEA2INkAACAbkGwAAJANSDYAAMgGJBsAAGQDkg0AALIByQYAANmAZAMAgGxAsgEAQDYg2QAAIBuQbAAAkA1INgAAyAYkGwAAZAOSDQAAsgHJBgAA2YBkAwCAbECyxWTx4sV9+/pZWZ3r0+cvdnbVdnbVTk4F/v7+AQEBCxYsUKvVarU6JOTtESPK9Xp4FKgNiYjYPHZs9dix1b6+Z/38NMHBxxINiYlJ9vPT+PlpgoKOv/jiZytXnjxgyK5d+QsXluslB+Tl5Wk0mqKi8uTkar0ZGZc0hpSWnv3ii6vl5U28lZWNDYY0Nzez/mUD0BNBskVj8eLFHMdxnD/H6Qg1nDE9YoCV1TT+LguLZguLpt69TzgbMmhQiIVFk6Xl9b59v9Wf20gmTIjiT3vkyW/GjBn6c1vXJ781a9boz2285Mlvy5YtQs5thYXnVq+uXb26dvv2bzo8t506dSkvrzkvr/nMmesdntv+9rdr+jNfeXlTTc0/WT9PgbxBskWjX79+EsklBsh9gI2NTXl5uU6nO77QJ2f6oIxJFrxHZjl98fZc8ll3fKFPzszBB6bYZvvbZfvbffqad/WBLeSAL96eW/DymGx/u+yp/fJmj/hi7dzaYxnkgAu73vr8jZl5s0fkh4z8OHzsxd1/atAcIwd8eyS5fPPrRdF+vF+lrDYaUP9FwTfZ753duJD3rx/vba4uJwf889sL9acLv81N5m26WPqvhlqjAf+o+LzxfGnz1+XNX5f/2Hj1/r9+IAf87z+bWq5evvnthf/9Z9MDz8xHCyRbNAYNGiTlCmCAvAbY2NjodLrcIOd0D460KNqffNa1H6BZr+6BA3oOSLZoTJ48WeIVwADpD7BWNIy0Xc/7i/bnDpK9yO8XrVZvxy375Re9HQ74hUAKwUWyhYNki4a/vz/HcRxnx3H+HOcfElKwfHn1n/50adcuzY4dp9555/CaNQfWrDmwYkV+YGC53pdeOsnfrjcycp+PzzEfn2Pjxn05dqzGx+eYWp1I+uKLyWPHasaO1Ywc+Zfhw6tGjiyYMUNNOmXK20OGlOslBzz77NyxY/1dXaNsbKr1DhxYMHasP6mLy1ILi+sKRRNvr14nHB2dSW1t50gtdo/KgP+obft3hqeFUaq+jA/Xtv1bb/sBpxMiHmhA5uReXQ/ImmLTfsAvP7fpPeBvZ9zTRIMBB58bYPwIiRHan9v0th+gSXz1F+3Peg9Pc2w/gPWMZwOSLRq/JftXdu3SnDmj67GePHk/P78hP78hNbUpJaVp795G/j/1Hj58LSWlaefOq2+88bX+3Ea6ZUv58uXVevkB779foj+3dX3yi47eoj+38ZInv/DwNULObaNGRfTrV9KvX4mtbVWH57bhw1+xsrpgZXVBobjW4bmtf/8Q/ZlPoWiysjr2QMk26lT7nvbYAUg2oMXbew7HOevdtu0s825CGflJ3v+Qve5t8YO27d8/NtbdvlLzfdUZ3h8b6+79cINs2Y+Ndc3fnv8u/4OanO01Oduvf3Xqf65+Rw64/tWp+lMf1eRs/zrz3bMbXvvbp5nfV2nIAX/7NPObrM1nN7ymSXy1dOULl9LW/f2Lj8kBl9LWlW1adGKRL2/7AacTIoqXz8gLfor3dELEXwv3kgNOLPIreMk928+O98Qiv6/3JxkN+CjsDxmTLPmrrLnPDzu/cyU5IPf5YQcDBu7zVCDZSLZo2NlVk1Nu+fJq5hWAMjJnbxP5/LG3aiCbBY38Rfsz6xnPBiRbNJBsSOOnebf0Fx5H2q4fafMm8yxK05++b/ixse5O09Wfvm/46fsG7c9trKe+WUGyReN3v6shkx0fX8+8AlBGfv5xs9GL3MzjKE0PBgwkX9RuuXqZ9dQ3K0i2aFhYGPyPbUpKE/MKQBn552O3kGwhZkwyWAPT/HV595PzEQLJFo1evS5xXIPe1FQkGz6An3xi8Fe2o6Mj8zhKUyQbiIOzszM55fLzG5hXAMrI/PwG8vnjPGwY8zhKUyQbiAOSDWlEsgWaOVlJJvvmtxdYT32zgmSLBpINadyf+l/kipGxDmuZx1Gant+58nRChCbxVc16tWa92ugNpB55kGzRQLIhjbs3XzJcl32NeRylLNZlA1qQbEhjctJFJBvJ7hYkWzScnLzIDetHjlxjXgEoI99fcwq7H5HsbkGyRQPrsiGNSDaSLQQkWzSQbEhjStKX5OXHp+1imWdRmv7YWEduWGc9780Nki0aCsUtMtl79zYyrwCUkalbislrIQ42SuZxlKZG78LKet6bmx73Az88jN7sGJcf4QOZ/v5Jg2Tb9mIeR2mKZANxsLBoIjesI9nwgcRWGiRbCD3uB354YJEfpBHJRrKF0ON+4IeHhYUFOeW+/LKNeQWgjESykWwh9Lgf+OHBGcI8AVBeJr97kVwxMn7wG8zjKE1PJ0SQG9ZZz3tzg2SLBpINaYxZUEhevn7Stpx5HKUs1mUDWpBsSOPyuR8ZJLsPkt11srWsZzwbkGzRIHerc5wz8wRAebloVhqZbOfeZ5hnUcoi2YAWo3XZzBMA5SWSjWQLAckWDSQb0hgXkUZefvR0iGaeRQna9n//77cN6/U/fd/wv/9sYj3vzc0jkuyamprS0lK2x4BkQxo3xGWQ10LcB9kx76MEvX2lhlzhd/CPA9nOevPziCTb19fX09OT7TEg2ZDGd1dlI9nd+n3VGTLZGZMs2M568/MoJLuxsZHjuK1bt7I9DHK3Osc1ME8AlJe7dmnIZPv5+THvowRFss2a7Jqami7uvXv37pUrV0x42KSkJIVCcf369S7GXLlypbi4uKioqKKiQvtwLlxgkR+kEclGsoVgvmS3tLRwHKdSqRISEoqLixsbG1tbW+/du1dWVpaTkzN37lylUpmWlmbCI7u6uj777LOd3ZuSkuLs7KxUKu3t7ZVKJcdxlpaWYWFhXZ8/TADJhjQi2UJs/rqcTHbm5F7izmLpY75k3759m+uSuLg4Ex62srKS47isrKz2d92/fz8gIMDT07OsrEx/Y2lpqaurK/8dN2/ebPrPY0hzczP5s/Tr58g8AVBern79E3LFiO8obFjvwHs/3CA3rF/Y9ZZYU1guSCLZ9vb2hw4dMu1hY2JilErl/fv3298VHBwcGBjY1tZmdHtLS4ubm5u41W5oMHhPH0dHbKWBD+aCgKPk5Wv3gfnM+yhlsS77ocMnOywszMHBge+aUqn09fVNSkpqaWkx7TG1Wu2AAQMWLFjQ/q6jR486OjrevXu3wy+8fPkyfwwKhaK2tta0706CZENKX3wmi0z2WMcC5lmUskj2Q4dPtv4/2//xawJFRUUcx3W4Itvf3z8sLKyysrKzr/X39+fzumTJEvojuXr1GrlbfeBAL+YJgPLy+ae3k8ke0z+HeRalLJL90DFKtiiEhYUNHjy4/e1arVahUPBFTkpK6vBr4+Pj+QFOTk70R1JebvBZvQoFPqsXPphINpItBBkn++7du5aWlh1etGxtbdW/RuHs7Nzhlx86dEg/hv5gkGxI6Xy/d8jLj889qWaeRQn673s/kRvW791upp+88kLGyc7MzOQ4rrq6usN7HR0d+RzPnDmzwwFksulXaiPZkNLoF1eSl0NmqJ5i3kcJWn/qI3KRX8HLYyhnruyQcbJ9fHzc3d07u7eqqio8PHzFihU3b97scID+hREHBwf6g0GyIaWL561Bsru1Jmc7mexsfzv6ySsvGCS7vr4+Li5OpVINHjzYwcFh+vTp6enpD/p37rVr1ziO27Fjh8nHo7/8GBkZafKD6LlwweDj1a2sLjFPAJSXa9YcIJP9akQE8z5KUCTb3MlOT09XqVRFRUX6RmdnZyuVSpVK1fWOcyM2btzY7Sb1LuDfloRHlG2QWOQHKUWykWwhmDvZQUFBra2tRnft3r2bv05469YtgY/m4uISEBBg8sHExsbyE2PZsmUmPwgJkg0pRbKF+F3+B2SyDwWI8KqmvDBfsu/duxcQENDhNkWtVstfLQwODhbyUBcvXuQ4Ljs727Qjqa2t5ZcAent7i7I8XKfTXbhwgZxvf/gD1mXDB1MdlE6uGJnj+xbzPkrQ/7n63a8b1terNevV1Qe2iDJ/ZYRU3nx1wYIFfOwqKiq6HbxixQqlUtn+r3UhaLVaT09PjuNUKpXJuy7bo9EYvKfP2LH+zBMA5WWAWx55Bdt3RBrzPkpZ3S+/iDV55YVUkp2SkiLwYiC/SV2tVpv2jWJiYjiO8/T0vH37tmmP0CFINqTUzyWbTHbAH/Yzz6KURbIZc/ToUT52He5mJDl+/DjHcadOnTLhu/BLuX19fe/cuWPSYXZKaelZcsO6m9sc5gmA8nLykBQy2X7DkplnUcoi2Q+d1tbWLlbyHTt2jE+2QqHo+nFCQ0O7zXqH8G9IEhoaatorKl2TnFxNzjcbm2rmCYDyEslGsoVgpmRHRkZyHGdra9vZS9X6ZHNd7kW8c+eOpaXl6tWrH/QAzp8/b21tHRsb+6BfKBAkG1I64w/LycuPYRMimWdRguo3rP90/e/YsP6waGtr0+e4s9eg9S+MWFpadvFQ6enpHMd99913D3QAtbW19vb2nb0/VFRU1AM9Wocg2ZDSoMnB5OWQRYG+zPsoQbFh3dzJ3rhxY4dj+BZzHOfh4dHFQ/n4+KhUqgf67jdu3HByctq2bVuH996/f9+0l1mMQLIhpcG+YUh2t2IrjZleGHF2dg4JCUlJSelsgFqt7rrpOp2uvr6ee8BN6vwH0OzZs6ezAUVFRSEhIcIfsDP27PkLuWHd3v5L5gmA8lKtTiSTnbBuHfM+StDqrM1ksg9O7Uc/eeWFmZK9evXqrj+ya/jw4RzH9e3bt4u1d/wm9c7e5qk9ra2tvr6+XXwEsFarjYqKEuUjDrDID1KKZAux4r1lZLKPBA6ln7zywkzJvn37trOzc2cf2ZWTk8M/TXNycrp4EBcXl87eSbVDQkJC+GueHWJtbc3vgez6mwqkpKSEnG+enjOYJwDKSyRbiOWbl5DJPvrS0/STV16Yb5FfSUmJo6Pj+fPnjW6vqKiwtbXlOG7r1q1dfDm/SV14XqOjozlhmLbE24gDBwzeIGLGDDXzBEB5Ocd/C7liZOHszcz7KEG/r9KQG9Zrj2XQT155YdatNMXFxfb29kFBQfn5+aWlpcePH1+yZIlCoRg6dGhxcXHXX7ts2TJra2uBS6rJC57dIsoybSQbUjrlKYOP650x5iDzPkrZX7Au2zzcuXNnx44dISEhXl5eM2fODA8Pz83N7fbNsrVarb29vShvbP2QQLIhpV5DU8lkvzAhj3kWpSySLWn4jTZlZWWsD6RT9u7NITesP/vscuYJgPJS1T+TTPaccduZZ1HKItmSJjg4WJTPQX94LFxYTs63IUPKmScAyks3+1zyKTRv0ibmWZSySLZ0aWlpsbS0XLt2LesD6QokG1Lq7bSYvPwYO/tN5lmUoP93p+XXT1i//vefvm+4/68fWE99cyODZKelpXEcd+XKFdYH0hVINqTUc4QHeTlkyxuvMe+jBP3GcCvN5288wKrfRwMZJNvb27vrXexSAMmGlHqN9kKyu/XshtfIZOfNHsF66psbqSe7rq6O47jk5GTWB9INy5ZpyA3rTz5ZwjwBUF7OmKEmk521P5N5HyUoki31ZPOb1IV/jC8rsMgPUopkC7HkjVlksj+JmMh66psbqSd79OjRQUFBrI+ie5BsSCmSLcTiZdMMXstePp311Dc3Uk/2qVOn6uvrWR9F9yDZkNLn3NeRK0YSY/BxvR349y8+JjesN2iOsZ765kbqyZYLSDakdIxDIXkFO3Im/sruSqzLBlRs2bKFTHZ4+BrmCYDycozDx0g2kt0tSLY4vP32JnLD+ty57zFPAJSXLn1KyGQvnBrPPItSlvWMZwaSLQ7BwcfI+TZiBD5IDD6Yw3qXkU+huNmJzLMoZVnPeGYg2eKAZENKxw80+IT1ncvwV3YH3m+59esnrH/f8NP3Da0//Yv11Dc3SLY4BAYWkcl2db3EPAFQXg7v70ReDjn64XvM+yhByzYtIhf5lW9+nfXUNzdItjj4+WnIZI8dq2GeACgvRzzhTCa7IA1vvtqBJxb5kskuivZjPfXNDZItDtOnl5Ab1j09sWEdPphjx/qTyT79xSnmfZSgSDaSLQ6JiQaftapWJzJPAJSXSDaSLQQkWxyQbEgpki1EbFhHssUByYaUThr2Nrli5NCeY8z7KEGxYR3JFgckG1LqbGvwlutb4j5i3kcpy3rGMwPJFgckG1LqbINkI9ndg2SLw+rVW8gN6+HhW5gnAMrLYb3PksmOj0hmnkUpy3rGMwPJFgesy4aUItnC/eXnNtYznhlItjgg2ZDScf2XkpcfP1y7g3kZJei9H2782Fh3p/Eqv2H93/fusp765gbJFgcfH4MXIj08TjFPAJSXw/o6kJdDivIOMu+jBC1ePoNc5PdN9nusp765QbLFwdX1PJns6dNPM08AlJdP9h9CJvuzj3KY91GC5gU/RSb77MaFrKe+uUGyxeHppw02rM+erWGeACgvsZUGyRYCki0OarWanG9r1hxgngAoL4cOHUU+hS7/13fM+yhBkWwkWxyQbEipo6Mz+RSqv1rHvI8S9GiYG5nsiq3LWE99c4NkiwOSDSl1H7CWXDFyqeIb5n2UoH8t3Hs6IUKT+OuG9ebqctZT39wg2eKAZENK7a2ukVewSwsvMu+jZMW6bEALkg0pRbKRbCEg2eIwd+5ycsP6ypU5zBMA5aW9VQOZ7BP7i5iXUboi2YCSESMMttIEBpYzTwCUl0g2ki0EJFsckGxIqVvfOPLyY9nxUvZllJ53bzb+2Fj3428b1n9uvc966psbJFscjJLt8LuvZg1PnzU8PVoVt3eW977QYL3rnnszaHResNsR9YTd6gm7V/mt+fWuF0OyFoRnvfb6e3M3R/nn6V0bnJL9+kq9OTHr9izeER/x6Tr1Z7w7lhfkrc/g/ejd7E92Fua//+fMrTV6c3Z9fSKzkvezQ9WnPmkoKriZn68rKGDfKajXTtmbvBzy7fkzzPsoQQtecicX+dWfLmQ99c0Nki0Ov3/iBJlsvd79D5DPsHQP7lVntVwGPP5YW3/ltf7KawGOu1In9CZd9NT8/sprA5X1I23LRtqWBTslpE3qkzrJfs9kJ95loxe49DmvN3joFv72tCnOab7DU59VrZyw0GvwCb3zR7+7J8AjddrE9ECffUFT9ec2vUsnb94XGpzx0pysV+Znvfpa9usrN764Y7ZH6ZyJJe3PbYeWrjoStyE5On3x7LPL5mjan9t4097K4e9KjPzzxqVlH649oz+38R7dU9Xhya/kkxuiJ9veug+S3a3ZfnbkE/Xb3GTWU9/cINniMKT3zt92q/8gkeBiAKsBe8c//uH4XrwLh8/rb3Wtv9U18uS3Z0KfDyY68EaOWNhP2TjYutbisTMcp3lCuSZuJBc3klv3tEXC0xYhLr52dtX9+tUMH35uxFNl48cfjZ4+ae14W70LPKaNeKpM7/jxR2ND/N/16veuV/9Ev2GJfsOWT3/Rz/e03uAXPol/bU6i37CEqU+umzVuQ9CEdxcuXRh5Vm9s7J9TE+J2zvXZMc9/17K5u5bN3f7WmlWrinnj3zm1L/1i0eH03IQleg+/v2PnjqqdO6oOHzyXm1Px8dGzf/7kaEXhoUtF+byVX5w6e+ba2TPX/vvK1fqrdfVX675vakSyTQPJFodpTkr9Aj9yDk97YpvRJA91eosc8JxDstGAl37/ZtcD5v5+OTlgqkOK0YD5QxeTA3wHpnU9wGdAhtGAV4ZFSbOGj/wAB6XBgOmO/uS9HKfx7s/1hAG+TyjSPbi08dx7Yx57b8xjM52mWFg0WVg09bVs6G/VMLFfHpINqNj1rJONBZ/s9eST7/lB642m6PODJDdg1qB3jQYED45/2ANChhic22Y4bjEaYHRuoz/5tT+30Z/8jM5t7U9+D5psO8uSuJH+cSP9pdxT5gNG2mr4289lbmU99c0Nki0OebNHJI/lIp/kXv792IhharVr3AKv3AVeudteXFsU7Ve4YHzh/HG8GwMWveS+N8Qta+rw41OHH1/tu/LXu8LHfjRnZO4LT8Z6LPR1Oqo31mNhziwnvQcDBkaNXjbKrnxU3zLehSNfz5pi86s+vTO9rF4eFtdf2aD3paEr93kqfnXi43zL+ls19LFsls5pQ6YDRD+36eXvfaa/cctmOBp8+ZQB3QzwHWg8YNoTBgP8HYwHPOdgMODZdgOmdjfAd6DBgIB2A3wGGAyY9kQ3A4yS7dhL86oz96ozt3vD26ynvrlBssUhb/YI8hlW+mbQzZpz8vVGlebvX358uejoydTDJ1MPX8zef7kglfRi9v6TqYc/25Wbvf7T7PWfln5woCpjw6W0dee2x/B+vHbDh0tS9H68dgN/+5kNr51et+DkWyGp82Pipu3Smzo/pvTNoM9XzDwRNeXTSG/9uU3vxoBFhfPHfTx3TH6IS17wUzmznN6ZFDbHZWfQUx+2P7cdnjE4+9l+bzy94JknciYNLOjg3DbFJmuKzcKRr+vPfK59zr721ML/nNs8Ffs8FS8NXdnhyS/d47GHdFYwSrZ3u2QHDTb4cpkOmDXIYED7M1N3A5o57gDHHfjgg5Osp765QbLF4chMJ/IZ9sXaecyzC1nJn/B4vzl+vOzI52cOniBPft/l7a7K2MCr2f3eia27D6/ZbZTsTZF/2hT5p8Q3spYurY6KOh8YWDZrVnlY2Mk/zY9ZM22G3qVz1s2aVa43LOzk0pDXlvtMXuE9caXnqJWeo+b5RKlUGr3PPHMs+o8vrPQc9daEEWvHD14zzjHo6dAhQ8r1jhxZMH+CR9yYAavc7de7917v3vuF0aH9+tXy2tldHjiwIHTU799xU+p9znmGjU21jU21UnndyqrZzu7Lea6OW8c8vnXMY7xTnpiqUDQpFE0c9zPH6aysLrw4TEkX/V9NTq5mPfXNDZItDtn+Bheyz22PYR4OKC+NYvTllz3akpL7R4405Bz4LnPr8cytx/fvrty1q2nXriZr6++QbCACSDakdKSthpR5NKWpUvkPMtkHD37FeuqbGyRbHJBsSKnRa9zM4yhNrayayWTn5V1gPfXNDZItDkg2pBTJFqKzcyLHqfV+/nkZ66lvbpBscUCyIY3/XXyQfP6keSiZx1GaqlT+5LZ+jUbDeuqbGyRbHJBsSOPlglTy+bNnQm/mcZSmRh9qXFtby3rqmxskWxyQbEgjki1Qow81bmhoYD31zQ2SLQ5INqTxckEqv0+d903XGczjKE2RbCRbHJBsSOPlglSsy0ayhYBki8P+KTZksitT1jCvAJSRSLZAe/fex29V5/3660bWU9/cINnikDHJgkz25YJU5hWAMvLS4Qyy18rH/5d5HKWpQtFE/qLKy5tYT31zg2SLA5INaTyZepgsUX/lNeZxlKZINpItDkg2pPFsRja5W92jXwHzOEpTJBvJFgckG9KIRX5ItkCQbHFAsiGNSLZA7eyWkBvWL1/+B+upb26QbHFAsiGNSLZAscgPyRYHJBvSiGQj2QJBssUByYY0fnvE4LOGUyfaMI+jNEWykWxxQLIhjedTN5Eb1le5Pcc8jtIUyUayxcHozY6vXyxlXgEoIwvj3zVcl93IPI7SdOBAJzLZTU1YMQJMwijZzBMA5eXHazcYJruJeRyl6WOPHSQ3rLe1tbGe+uYGyRYHJBvSePitrWSyB//ub8zjKE2N3omlB9Ijf+iHAJINafxwSQpZIpc+55nHUZoi2T3yh34IINmQxuyV75Mb1qcOymAeR2mKZPfIH/ohgGRDGs9tjyGfPx9OdmIeR2mKZPfIH/ohgGRDGpFswclWk7Ke9wxAssUByYY0ItmCk20A63nPgJ74Mz8MkGxII5KNZAukJ/7MDwMkG9KIZCPZAumJP/PDAMmGNJa+u4bcsL5mzHTmcZSmSHZP/JkfBkg2pBHrspFsgfTEn/lhgGRDGpFsIX7yiY7crc5x21jPewYg2SJwu66G7PW+CY8zTwCUl1vm7yKT7d7vS+Z9lKCpqbfI35JC0ch66jMAyRaB5q/LDZLtqWCeACgvV72QTsZo8hMfM++jBN21y+CDHy0setzb+OmQbFFAsiGlm0I3khvWg5/azbyPEhTJ1iHZooBkQ0pL3wwyWOQXMIF5HyUokq1DskUByYaUItlC3L//H+Ru9YEDl7Ce+gxAskUAyYaUItlCPHKkgVzh5+zszHrqMwDJFgEkG1KKZCPZAkGyRQDJhpQi2Ui2QJBsEUCyIaX5iyLIDevrJr/IvI8SFMnWIdmigGRDSuOmGWyl8RpcxLyPEhTJ1iHZooBkQ0qRbCHu3dtIbljv23cf66nPACRbBJBsSCmSLcQNG74mfwtCciUAABsdSURBVEtK5Q3WU58BSLYIINmQUiRbiPHxF8jfUq9ezaynPgOQbBFAsiGlm2YnkhvWXxjxIfM+StCYmL+Qye7T5zLrqc8AJFsEkGxIKRb5CXHp0moy2XZ21aynPgOQbBFAsiGlSLYQN2w4R25YHzNmFeupzwAkWwT+diyDnG+Zz1gzTwCUl5+9PpV8CqXN9GbeRwm6c6eGXOTn7+/PeuozAMkWgW9zk8n5ljXFhnkCoLw8ETWFfArtDfRh3kcJimTrkGxRQLIhpZ+8MsEg2bMDmfdRgiLZOiRbFJBsSGna7EByw/qmGdHM+yhBkWwdki0KSDak9CX3veRaiOdH5zPvowRFsnVItigg2ZBSJFuIcXFl5Ib1kSOzWE99BiDZIoBkQ0qRbCHOnVtC/pYcHWtZT30GINkigGRDSpFsJFsgSLYIINmQ0uWTDTasv+rxAfM+StCgoDNksp988iLrqc8AJFsEkGxIaeH8cQaL/ObMZt5HCTprVjmZ7BEjyllPfQYg2SKAZENKkWwhLlpUSG5YnzlzE+upzwAkWwSQbEgpki3EVasOkIv81Go166nPACRbBJBsSCmSjWQLBMkWASQbUno0dJTBO4vNm8e8jxIUydYh2aKAZENKdwZMJzes7wxdxryPEhTJ1iHZooBkQ0p9nY6SayEifXOY91GCItk6JFsUzm1bQSb70B+fYJ4AKC+Nkr3QD1tpOjA8/HNyw7q3NzasA5M4u3EhmeycWU7MEwDlJZItRKzL1iHZooBkQ0qRbCRbIEi2CCDZkNKXXbeQG9ZjZ2Qw76MEnThRQybby0vDeuozAMkWASQbUpozy8ngCvaiOOZ9lKAqlUGy/fw0rKc+A5BsEUCyIaVIthCffz6N3LD++utprKc+A5BsEUCyIaVIthAjIhLJRX6JiYmspz4DkGwRQLIhpUg2ki0QJFsEkGxI6cGAgeRT6PAbCcz7KEGRbB2SLQpINqR0i9ckcsP6nuidzPsoQZFsHZItCkg2pNTniRxyLcTrz3/JvI8SFMnWySXZWq02LS0tNDQ0MDAwPDz81KlT+rtqamrUanVgYGBoaGhaGpsryEg2pPQZo2QHn2XeRwk6dWoeuWH95ZfzmMx3tsgg2Tdu3PDy8kpISKivr79z505sbCzHcUlJSTqdLj8/39fXt7KysrW1tbi42MHBITIy0vxHiGRDSicMOEYmO+ZF/JXdgViXrZN+su/evevh4VFWVkbe6OLiwnFcbm6um5vb3bt3dTrdlStXbG1tOY5TKBTmP0gkG1I6qm8ZGaP4iGLmfZSgSLZO+smOjo5u/3LHggULOI6ztLTMycnhb8nKyuJf3nJycjL7MSLZkNY/DvmQ3LC+OfoY8z5KUCRbJ/Fk19TUqFSq9reHhIRwHGdtba3Vavlb6urqXF1dVSpVRUWFeY9Rp0OyIbVZU2zIp9CRhEzmfZSgSLZO4smOiorKzMxsf3vfvn05jgsLCzP/IXXI58unk/Pt6EtPM08AlJdIthCxYV0n8WSfOXPmzp07Rjdev36dfw0kJSWFyVG1pyjaj5xvhfPHMU8AlJdIthCxyE8n8WR3SH5+Pv8PVlVVxfpYfgXJhpRm+fQmn0L5G7OZ91GCItk6OSZbrVbzL2SzPpD/gGRDSjO9rMin0MfbC5n3UYIGBS0mk81qHwZb5JdsfoVfaGgo6wP5D0g2pDTBfQK5Yf3wllLmfZSg06eryWQfOHCA9dRngMySffPmTf5fKzk5mfWx/AckG1Lq2ucMuRYi6a1a5n2UoEi2TnbJ1r+QXVlZaXRXS0tLbGwsk6NCsiGlSLYQ3dyyyA3ra9d+zmS+s0XSya6rqzt9+jR5S3h4OMdxSqVS+9uKbD0pKSlMdqvrkGxI7bDeVWSyt79zmXkfJeiQIQYf17twIT6uV0pkZ2crFAqO4zZv3szfotVq7e3tOY7z8vIyGtzW1ubi4lJbW2v2w9TpkGxIbX9lAxmjfVtrmPdRgiLZOikne/To0fxrIPHx8fwtmZmZTk5OHMdNnz7daHBsbGxCQoLZj/FXkGxI6cT++eSG9QPbq5n3UYIi2TopJ9vb25vjOA8Pj1u3bul0urq6utGjR1+/fn3cuHFKpfL27dv6kUlJSYGBgeyOFMmGtO7zVJBPoU8zKpn3UYIi2TopJ7uqqmro0KGRkZHZ2dmrV68eN25cTU2NTqe7ceOGu7u7m5tbSkrK5s2bx40bFxcXp2330rY5QbIhpUi2ED09N5Eb1t97r5DhrGeFdJOt0+na2tqKiopycnJKSkqMolxRUZGbm1tYWHjz5k1Wh6cHyYaUItlCxCI/ncSTLRcKwv5AzrfPlv2ReQKgvESykWyBINkikDd7BDnfSt8MYp4AKC+RbCRbIEi2CCDZkNI1f/AmN6wfTa1i3kcJimTrkGxRQLIhpUbrsrNTGpj3UYJOnDiDTHZJSQnrqc8AJFsEkGxIaX8rg2QfTG1i3kcJOmjQPnLD+sGDX7Ge+gxAskUAyYaUGiU759D/Me+jBLWxqSZ/S8nJ1aynPgOQbBFAsiGlRsk+coR9HyUokq1DskUByYaUTuyXR25YLyhg30cJimTrkGxRQLIhpeTzJ92DYx5HaYpk65BsUUCyIaVIthCHD19FblgvLDzHeuozAMkWASQbUopkC1Gl8icX+Wk0GtZTnwFItggcDBhAzrezSYuYJwDKSyQbyRYIki0C2f525Hw7tz2GeQKgvESykWyBINkigGRDSsnd6nEj/ZnHUZoi2TokWxSQbEgpuRCC49jHUZoOHTqKTDarDw5kC5ItAkg2pBTJFqKt7XvkhvUvvrjOeuozAMkWASQbUopkC9HK6gL5W8rLa2Y99RmAZIsAkg0pRbKRbIEg2SKAZENKyd3qI201zOMoTS0srpLJxgsjwESQbEjjtbITWOQnRIWiiUx2eXkT66nPACRbBJBsSOPfv/wYyRaind0ScsP65cv/YD31GYBki8D+KTbkfKtMWcO8AlBG1n6aST5/0jx6MY+jNHV0dCYX+TU0NLCe+gxAskUgY5IFOeUuF6QyrwCUkZcLUsnnz54JvZnHUZoi2TokWxSQbEgjko1kCwfJFgEkG9L4dW46uVv9rdFTmcdRmvbr50gmu7kZi/yASSDZkMaTqYfJhRD9ldeYx1Gacoawnvds6KE/trgg2ZBGJFtwsreRG9Z/+IH1zGcBki0CSDakEckWnGyDDzXukS9lI9ligGRDGpFsJFs4SLYIINmQxrMZ2eRudY9+BczjKE057j6Z7Pv3Wc98FiDZIoBkQxqxyE9wsg3smfTUn1tUkGxII5ItONlqUtbzng1Itggg2ZBGJFtwsrHID8kWAyQb0ohkI9nC6aE/trgg2ZDG6qzN5PPng4kDmMdRmiLZOiRbFIzeOfP6xVLmFYAy8tTmeHLD+hr3aczjKE2RbB2SLQpGyWaeACgvP1ySQi6EcOlznnkcpSmSrUOyRQHJhjQi2YKTfYCU9bxnA5ItAkg2pBHJFmJBQTPWZeuQbFFAsiGNSLYQjxxpQLJ1SLYoINmQxuyV75Mb1qcOymDeRwm6Z08j2WuF4hbrec8GJFsEkGxI47ntMeTz58PJTsz7KEF37TL4eHULi5748eo6JFsUkGxII5ItxP37/0HuVh84cAnrec8GJFsEkGxII5ItxCNHGsgVfs7OzqznPRuQbBFAsiGNSDaSLRwkWwSQbEjjmQ2vGWxYf8aVeR8lKJLNg2SLAJINaTzxp2hyw/qq8aHM+yhBkWweJFsEkGxI486ITeRaiBF9K5n3UYIi2TxItggg2ZBGJFuIe/c2krvV+/bdx3reswHJFgEkG9K4ae4OMtlu/c4w76MExbpsHiRbBJBsSOOqF9LJGE164hjzPkpQJJsHyabldl0N2et9Ex5nngAoLzeFbiA3rL/w1AfM+yhBkWweJJuW5q/LDZLtqWCeACgvS98MMliXHTCBeR8lKJLNg2TTgmRDSpFsIWLDOg+STQuSDSlFsoWYknKBXOTn5eXFeuqzAcmmBcmGlH6+Yib5FNrzRy/mfZSgO3dqyGT7+/uznvpsQLJpQbIhpSeiphj8lT0zgHkfJSiSzYNk04JkQ0oPzg8mN6yv832NeR8lKJLNg2TTgmRDSmN8N5NrIbyGlDLvowRFsnmQbFqQbEgpki3ENWu+IjesDx+ODevAJJBsSOnr3slksv2Gfcq8jxJ06dJq8rdkZ1fNeuqzAcmmBcmGlM4bn0XGaNaIQ8z7KEGRbB4kmxYkG1Ia47uZ3LC+YMz7zPsoQZFsHiSbFiQbUopFfki2cJBsWv52LIOcb5nPWDNPAJSXn7wygXwKpc0OYt5HCbphwzlyw/qYMatYT302INm0fJubTM63rCk2zBMA5WXh/HHkU2jvnNnM+yhBsciPB8mmBcmGlCLZSLZwkGxakGxIKZKNZAsHyaYFyYaUps0OJDesb5oRzbyPEhTJ5kGyaUGyIaVhoz8k10IEPo0PEkOyOwXJpgXJhpS+OOoDMtnPP32CeR8l6KuvfkFuWJ84MZ311GcDkk0Lkg0pRbKFOGtWOflbGjGinPXUZwOSTQuSDSlFspFs4SDZtCDZkNIlHqvJDeuvTspi3kcJimTzINm0INmQ0vwQF4NFfvOimPdRgiLZPEg2Lee2rSDn26E/PsE8AVBeHgkcSj6F9kfGMO+jBF20qJDcsD5z5ibWU58NSDYtZzcuJOdbziwn5gmA8jJnlpPB/6gtimPeRwm6atUBcpGfWq1mPfXZgGTTgmRDSpFsJFs4SDYtSDakFMlGsoWDZNOCZENK3/f1JzesJ89fz7yPEhTJ5kGyaUGyIaW+TkfJtRAL/fKZ91GCItk8SDYtSDakFMkWYnj45+SGdW/vLNZTnw1INi1INqQUyRYi1mXzINm0INmQUiQbyRYOkk0Lkg0pfdl1C7lhPXZGBvM+SlBf3yoy2e7un7Ge+mxAsmn54p15ZLLzgp5kngAoLw8+1598CmW/uZV5HyXoM88YfML62LGfsp76bECyaSmK9iPnW+H8ccwTAOVl1hQb8il0JCGTeR8laEhIFrlh/bXXdrOe+mxAsmlBsiGlSLYQIyISyUV+iYmJrKc+G5BsWpBsSCmSjWQLB8mmBcmGlCLZSLZwkGxakGxI6frx3uSG9YxVh5j3UYIi2TxINi1INqR0VN8yci1EfEQx8z5KUCSbB8mmpSjab8lT3O9/x/W3+k3lf3TpoxjVV5AeA618nXp36+xRA19yd+zMV8b/Pm7a2C5cGzzpwyWh3ZoVF3Ey9V3hnsv5gHn45CuSLcSpU/PIDesvv5zHeuqzAcmmZUfg6Mcf44BA+lganNKMdLR+rOsTm3s/SyEntoBhtl2c2Ew41Zl25hN4ChxhexbJ7laVSkP+lvz8NKynPhuQbFrCRtmxziCQOwYx+k0Nx3EKhcXAgU6Ojs6Ojs79+s3l71IomhWKpl69TvC363VyesXKqtnKqrl3729tbKoHDixQqfxJR46MsrGptrGptrc/OWDAyZEjC6ZPV5P6+Lw9ZEi5XnJAUNDiiIjEOXOSVSqN3meeORYRkUgaFrZt2rTTs2aV84aFnVy16gDpm28emzevdt682sWLv1m6tDou7tLOnRrS99+/FB/fHB/fvHPn9V27mvbsaTxypOHIkYbRo0uQbB2STc/62V6sJzyQO8kcp+G46vbJNsS/Zw/IIG8PCjrOeuqzAckWgelDf4fXRgA1/hLOpRQGHCBvnzYtl/W8ZwOSLQ61ZX8+d3R/5WcFRh7euHJ//DIhvr9ozpoQ325VTx45z2NYZ4a6D/Yb2qcLJw/+3Sg7i24dbqvor3xMuH0s28078MD4SziXUhjgQ25YDwt7m/WkZwOSDczKX89/2f7EpldzJL3rE1tKTISQE1vMzIldnNhMONWZduYTeAq0sWBeQ3kN4DiOKy/Hm68CACRGW1tbU1NTgyEXLjSVlzdVVjYa3f7VVzV5eRfy8i588MHXycnVGRmXNIYUFZUnJ1cnJ1evXfvXlSu/Xbny5AFDdu3KX7iwXC85IC0tLTExMSYm2c9Pozc4+FiiIcuWpbi6nh8xopzXw6NAbcisWSsdHUscHUvs7avs7KqdnAr8DXF3f6VXrwu9el2wtLxmYdHUu/cJ59/o37+/lZWVjY1NQkIC638ZZiDZAAAgG5BsAACQDUg2AADIBiQbAABkA5INAACyAckGAADZgGQDAIBsQLIBAEA2INkAACAbkGwAAJANSDYAAMgGJBsAAGQDkg0AALIByQYAANmAZAMAgGxAsgEAgqipqSktLWV9FD0dJBsAIAhfX19PT0/WR9HTQbIBAN3T2NjIcdzWrVtZH0hPB8kG4JGipqami3vv3r175coVEx42KSlJoVBcv369izFXrlwpLi4uKiqqqKjQarUmfBfQLUg2AI8OLS0tHMepVKqEhITi4uLGxsbW1tZ79+6VlZXl5OTMnTtXqVSmpaWZ8Miurq7PPvtsZ/empKQ4OzsrlUp7e3ulUslxnKWlZVhYWNfnD2ACSDYAjw63b9/u8PPI9cTFxZnwsJWVlRzHZWVltb/r/v37AQEBnp6eZWVl+htLS0tdXV3577h582bTfx7QDiQbgEeHLpJtb29/6NAh0x42JiZGqVTev3+//V3BwcGBgYFtbW1Gt7e0tLi5uaHaooNkA/DowCc7LCzMwcGBz6VSqfT19U1KSmppaTHtMbVa7YABAxYsWND+rqNHjzo6Ot69e7fDL7x8+TJ/DAqFora21rTvDoxAsgF4dOCTrf/P9n/8mkBRURHHcR2uyPb39w8LC6usrOzsa/39/flqL1myhP5IgA7JBuBRwijZohAWFjZ48OD2t2u1WoVCwRc5KSmpw6+Nj4/nBzg5OYl7VD0WJBuARwfRk3337l1LS8sOL1q2trbqXyh3dnbu8MsPHTqkHyPiUfVk8HsE4NFB9GRnZmZyHFddXd3hvY6OjnyOZ86c2eEAMtlYqS0KSDYAjw6iJ9vHx8fd3b2ze6uqqsLDw1esWHHz5s0OB+hfGHFwcBDxqHoySDagRavVpqWlhYaGBgYGhoeHnzp1Sn9XTU2NWq0ODAwMDQ01bQcHeCD0ya6vr4+Li1OpVIMHD3ZwcJg+fXp6evqD/p177do1juN27Nhh8vHoLz9GRkaa/CCABMkGVNy4ccPLyyshIaG+vv7OnTuxsbH6i1H5+fm+vr6VlZWtra3FxcUODg6Ytw8bPtnp6ekqlaqoqEjf6OzsbKVSqVKput5xbsTGjRu73aTeBfzbkvBgG6RYINnAdO7evevh4UFue9PpdC4uLhzH5ebmurm58St2r1y5Ymtry6/PZXSkPQU+2UFBQa2trUZ37d69m79OeOvWLYGP5uLiEhAQYPLB8OdvjuOWLVtm8oMAI5BsYDrR0dHtX+5YsGABx3GWlpY5OTn8LVlZWVjpZR7u3bsXEBDQ4TZFrVbLXy0MDg4W8lAXL17kOC47O9u0I6mtreWXAHp7e4uyPBzwINnARGpqalQqVfvbQ0JCOI6ztrbW/195XV2dq6urSqWqqKgw7zECA/izKcdxQv4hVqxYoVQq2/+1LgStVuvp6clxnEqlMnnXJegQJBuYSFRUVGZmZvvb+/bty++ZNv8hga5JSUkReDGQ36SuVqtN+0YxMTEcx3l6et6+fdu0RwCdgWQDEzlz5sydO3eMbrx+/TofhZSUFCZHBbrg6NGj/L9Oh7sZSY4fP85xHLn4Rzj8Um5fX9/2Tw9AD5INxCQ/P5+PQlVVFetj6Ym0trZ2sZLv2LFj+vdp6vpxQkNDu816h/BvSBIaGmraKyqgW5BsICZqtZp/IZv1gfREIiMjOY6ztbXt7KVqfbK73ot4584dS0vL1atXP+gBnD9/3traOjY29kG/EAgHyQZiwq/wCw0NZX0gPY62tjZ9jjt7DVr/woilpWUXD5Wens5x3HffffdAB1BbW2tvb9/Z+0NFRUU90KOBzkCygWjcvHmTL0JycjLrY+lxkMneuHFjh2P4FnMc5+Hh0cVD+fj4dLgWqAtu3Ljh5OS0bdu2Du+9f/++aS+zgPYg2UA09C9kt38D5ZaWFvz/8sPG2dk5JCSkiwu//MtWXTRdp9PV19c/6CZ1/gNo9uzZ09mAoqKikJAQ4Q8IugDJBiZSV1d3+vRp8pbw8HCO45RKZfvXSVNSUrBb/WGzevXqrj+ya/jw4RzH9e3bt4u1d/wm9c7e5qk9ra2tvr6+XbyBjFarjYqKwkcciAWSDUwhOzub39umb4RWq7W3t+c4zsvLy2hwW1ubi4sLPkrqYXP79m1nZ+fOfs85OTn8n9j6Xakd4uLi0tk7qXYIv3PKthOsra3550nX3xQIB8kGpjB69Gh+/sfHx/O3ZGZmOjk5cRw3ffp0o8GxsbEJCQlmP8aeSElJiaOj4/nz541ur6io4N/mZevWrV18Ob9JXXheo6OjOWGYtsQbtAfJBqbg7e3NX8Xi32Oorq5u9OjR169fHzdunFKpJP+/OykpKTAwkN2R9jiKi4vt7e2DgoLy8/NLS0uPHz++ZMkShUIxdOjQ4uLirr922bJl1tbWApdUkxc8uwXLtMUCyQamUFVVNXTo0MjIyOzs7NWrV48bN45/d80bN264u7u7ubmlpKRs3rx53LhxcXFx+DgSM3Pnzp0dO3aEhIR4eXnNnDkzPDw8Nze3238F/qUtXHKQOEg2MJG2traioqKcnJySkhKjHFRUVOTm5hYWFgq/igWYw2+0MXorXSA1kGwAgE6n0wUHB+PdcaUPkg0A0LW0tFhaWq5du5b1gYBuQLIBALq0tDSO465cucL6QEA3INkAAJ23t3fXu9iBRECyAejp1NXV4Z1h5AKSDUBPh9+kLvxjfAFDkGwAejqjR48OCgpifRRAEEg2AD2dU6dO1dfXsz4KIAgkGwAAZAOSDQAAsgHJBgAA2YBkAwCAbECyAQBANiDZAAAgG5BsAACQDUg2AADIBiQbAABkA5INAACyAckGAADZgGQDAIBsQLIBAEA2INkAACAbkGwAAJANSDYAAMgGJBsAAGQDkg0AALIByQYAANmAZAMAgGxAsgEAQDYg2QAAIBuQbAAAkA1INgAAyAYkGwAAZAOSDQAAsgHJBgAA2YBkAwCAbECyAQBANiDZAAAgG5BsAACQDUg2AADIBiQbAABkA5INAACyAckGAADZgGQDAIBsQLIBAEA2INkAACAbkGwAAJANSDYAAMgGJBsAAGQDkg0AALIByQYAANnw/wFluOGrgD1NKQAAAABJRU5ErkJggg==)
Example: $x^2 + 5x +6 = 0$. ["Completing the square."]
$(x+5/2)^2 - 1/4 = 0$
$[(x+5/2) + 1/2] [(x+5/2) - 1/2] = 0$
$(x+3)(x+2)=0$
...
Solution: $x = -3, x = -2$
- Cubic:$a_3x^3 +a_2x^2 + a_1 x+ a_0= 0$
Example: $x^3 - 5x^2 +6x = 0$.
$x (x^2 - 5x +6) = 0$.
... $x(x-2)(x-3)=0$
...
Solution: $x = 3, x = 2, x=0$
V. Visualizing Solving Equations. *
- Polynomial Functions: $f(x) = a_nx^n+ ... +a_2x^2 + a_1 x+ a_0$ with real (or complex) coefficients.
- Visualizing functions with tables and graphs.
- Visualizing functions with tables and mapping diagrams.
- Solving an equation $f(x) = 0$ visualized with table and graph. [Look for graph of $f$ to cross X-axis.]
- Solving an equation $f(x) = 0$ visualized with table and mapping diagram. [Look for arrow on diagram to hit $0$ on target axis.]
- Main issues:
- How do these visualizations connect to the algebra for solving an equation?
- How do these visualizations connect to having complex numbers for solutions?
VI. Solving equations visualized with mapping diagrams *
- Linear and Quadratic Mapping Diagram Examples with GeoGebra
Geogebra File to download from Geogebra
- Example: $2x +1 = 5$.
- Example: $x^2 - 5x +6 = 0$.
$(x-5/2)^2 - 1/4 = 0$
- Example: $2(x-3)^2 + 1 = 9$
VII. Complex Number Solutions to Real Polynomial Equations Visualized with Mapping Diagrams
- Quadratic Example: $x^2+Bx+C = 0$ *
- Solutions: $x = \frac {-B \pm \sqrt{B^2-4C}}{2}$
- Note connection of roots to value of (quadratic) discriminant $\Delta = B^2-4C$
- Particular examples to examine:
- Example: $x^2 -5x + 6= 0$
- Example: $x^2 -4x + 6= 0$
- Geogebra File to download from Geogebra
- Cubic Example: $x^3 +Bx^2+Cx+D=0$
- Replace $x$ by $x − B/3$:
$(x − B/3)^3 +B(x − B/3)^2+C(x − B/3)+D=0$
so the coefficient of $x^2$ in the resulting equation is $0$.
This changes the cubic to the so-called depressed cubic form $x^3 + px + q = 0$.
- Example: $x^3 - 5x^2 +6x = 0$; $B = -5, C=6$
Replace $x$ by $x + 5/3$:
$(x + 5/3)^3 -5(x + 5/3)^2+6(x + 5/3)=0$
$x^3 + 3\cdot 5/3 x^2 + 3\cdot 25/9x + 125/27 -5 x^2 - 50/3x - 125/9 +6x + 10=0$
$x^3 - 7 / 3 \cdot x + 20 / 27 =0$ ; $p=-7/3; q=20/27$.
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Solutions: Check Wikipedia: https://en.wikipedia.org/wiki/Cubic_function
- Note connection of roots to value of the related (cubic) discriminant
$\Delta = -(\frac q2)^2-(\frac p3)^3$
- Particular examples to explore: *
- Example: $x^3 - 1= 0$
- Example: $x^3 - x + 1= 0$
- Example: $x^3 -6x + 4= 0$
- Example: $x^3 -3x + 4= 0$
The End
Thanks!
Questions?