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% Document Info
\author{liv.skreka}
\pdfinfo{
  /Title (chapter-6-polynomials.pdf)
  /Creator (Cheatography)
  /Author (liv.skreka)
  /Subject (Chapter 6 Polynomials Cheat Sheet)
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\noindent
\begin{multicols}{3}
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    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Chapter 6 Polynomials Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{liv.skreka} via \textcolor{DarkBackground}{\uline{cheatography.com/201997/cs/42819/}}}
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\noindent
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}liv.skreka \\
  \uline{cheatography.com/liv-skreka} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 30th March, 2024.\\
   Updated 24th March, 2024.\\
   Page {\thepage} of \pageref{LastPage}.
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  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\end{multicols}}




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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6A The Language of Polynomials}}  \tn
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\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711249518_Screenshot 2024-03-24 at 1.54.20 pm copy.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\mymulticolumn{1}{x{5.377cm}}{- A polynomial function is a function that can be written in the form:  \newline P(x) = ax\textasciicircum{}n + ax\textasciicircum{}n-1 + ... + ax + a \newline  \newline - The leading term, ax\textasciicircum{}n, of a polynomial is the term of the highest index among those terms with a non-zero coefficient.}  \tn 
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{The function f(x)=x\textasciicircum{}1/3}}  \tn
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\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711254417_Screenshot 2024-03-24 at 3.26.10 pm.jpeg}}} \tn 
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\mymulticolumn{1}{x{5.377cm}}{Inverse functions \newline f(x)=x\textasciicircum{}1/3 is the inverse of f(x)= x\textasciicircum{}3}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6E - Cubic Functions of f(x) = a(x-h)\textasciicircum{}3 + k}}  \tn
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\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711254185_Screenshot 2024-03-24 at 3.22.20 pm.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\begin{tabularx}{5.377cm}{x{1.33664 cm} x{1.00248 cm} x{1.42018 cm} p{0.4177 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{5.377cm}}{\bf\textcolor{white}{6E - Cubic Functions of f(x) = a(x-h)\textasciicircum{}3 + k}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Point of Inflection (POI)}} & {\bf{Vertical Translations}} & {\bf{Horizontal Translations}} &  \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
A point of zero gradient & by adding or \seqsplit{subtracting} a constant term to y-x\textasciicircum{}3, the graph moves either up or down & The graph of y=(x-h)\textasciicircum{}3 is moved h units to the right (for h\textgreater{}0) &  \tn 
% Row Count 13 (+ 10)
% Row 2
\SetRowColor{LightBackground}
The 'flat point' of the graph & e.g. y=x\textasciicircum{}3 + k moves the graph k units up (for k\textgreater{}0) & the POI is at (h,0) &  \tn 
% Row Count 19 (+ 6)
% Row 3
\SetRowColor{white}
The POI of graph of y=x\textasciicircum{}3 is at (0,0) & The POI becomes (0,k) & In this case, the graph of y=x\textasciicircum{}3 is translated h units in the positive direction of the x-axis. &  \tn 
% Row Count 27 (+ 8)
\hhline{>{\arrayrulecolor{DarkBackground}}----}
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\mymulticolumn{4}{x{5.377cm}}{The implied domain of all cubics is R and the range is also R}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}----}
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\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6D - Solving Cubic Equations}}  \tn
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\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711253707_Screenshot 2024-03-24 at 3.12.53 pm.jpeg}}} \tn 
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\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6D - Solving Cubic Equations}}  \tn
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\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711253663_Screenshot 2024-03-24 at 3.12.39 pm.jpeg}}} \tn 
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\mymulticolumn{1}{x{5.377cm}}{In order to solve a cubic equation, the first step is often to factorise. \newline Factorise by identifying a factor, then using polynomial division.  \newline Factor the quadratic factor \newline Then, use null factor law.}  \tn 
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6B - Division of Polynomials}}  \tn
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\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711249811_Screenshot 2024-03-24 at 2.09.05 pm.jpeg}}} \tn 
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\mymulticolumn{1}{x{5.377cm}}{- When we divide the polynomial P(x) by the polynomial D(x) we obtain two polynomials, Q(x) the quotient and R(x) the remainder, such that \newline P(x) = D(x)Q(x) + R(x) \newline and either R(x) =0 or R9x) has degree less than D(x) \newline Here P(x) is the dividend and D(x) is the divisor}  \tn 
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\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711250235_Screenshot 2024-03-24 at 2.16.40 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Dividing Polynomials involving fractions}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6B - Division of Polynomials}}  \tn
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\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711249939_Screenshot 2024-03-24 at 2.11.29 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Dividing polynomials when we have a remainder}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6B - Division of Polynomials}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711273312_Screenshot 2024-03-24 at 8.40.24 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Equating Coefficients Methods instead of dividing}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6F - Graphs of factorised cubic functions}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711254587_Screenshot 2024-03-24 at 3.12.39 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6F - Graphs of factorised cubic functions}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711254708_Screenshot 2024-03-24 at 3.30.32 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Repeated roots/factors}  \tn 
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\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6F - Graphs of factorised cubic functions}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711254806_Screenshot 2024-03-24 at 3.30.43 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Cubic equations with one x intercept}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6G - Solving Cubic Inequalities}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711254903_Screenshot 2024-03-24 at 3.34.29 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\begin{tabularx}{5.377cm}{p{0.45947 cm} x{1.04425 cm} x{1.37841 cm} x{1.29487 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{5.377cm}}{\bf\textcolor{white}{6H - Families of cubic polynomial functions}}  \tn
% Row 0
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y=ax\textasciicircum{}3 & y=a(x-h)\textasciicircum{}3 + k & \seqsplit{y=a(x-a)(x-b)(x-c)} & y=ax\textasciicircum{}3+bx\textasciicircum{}2+cx+d \tn 
% Row Count 2 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}----}
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\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6H - Families of cubic polynomial functions}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711255285_Screenshot 2024-03-24 at 3.38.59 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6H - Families of cubic polynomial functions}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711255331_Screenshot 2024-03-24 at 3.40.32 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Quartic Functions}  \tn 
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6B - Division of Polynomials}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711273587_Screenshot 2024-03-24 at 8.40.24 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Equating Coefficients Methods instead of dividing}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6C - Special Cases Differences of Cubes}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711253273_Screenshot 2024-03-24 at 3.06.42 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6C - Rational Root Theorem}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711253117_Screenshot 2024-03-24 at 2.57.01 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6C - Rational Root Theorem}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{P(x) = 2x\textasciicircum{}3 - x\textasciicircum{}2 - x - 3} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Choose factors of -3, which are ±1 and ±3.} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{However,  P(1) ≠ 0, P(-1) ≠ 0, P(3) ≠ 0, and P(-3) ≠ 0} \tn 
% Row Count 4 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Therefore, we must use the Rational Root Theorem.} \tn 
% Row Count 5 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{We must use  P(± factors of constant/factors of leading coefficient)} \tn 
% Row Count 7 (+ 2)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{factors of constant (of -3) = ±1, ±3 factors leading coefficient (of 2) = ±1, ±2} \tn 
% Row Count 9 (+ 2)
% Row 6
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\mymulticolumn{1}{x{5.377cm}}{e.g. P(±3/2), P(±1/2)} \tn 
% Row Count 10 (+ 1)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{We now have to check these factors -{}-\textgreater{} P(3/2) = 2(3/2)\textasciicircum{}3 - (3/2)\textasciicircum{}2 -(3/2) -3 = 0} \tn 
% Row Count 12 (+ 2)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{therefore, x-3/2, which equates to 2x-3 as a factor} \tn 
% Row Count 14 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6C - Factorisation of Polynomials}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711251912_Screenshot 2024-03-24 at 2.44.24 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{For a polynomial P(x) \newline - If P(a) = 0, then x-a is a factor of P(x) \newline - Conversely, if x-a is a factor of P(x), then P(a) =0}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6C - Factorisation of Polynomials}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711251698_Screenshot 2024-03-24 at 2.40.55 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Remainder Theorem: \newline When P(x) is divided by bx+a, the remainder is P(-a/b).  \newline  \newline For example, if P(x) is divided by x-1, let x-1=0, x=1.  \newline P(1) = Remainder (R(x)) \newline  \newline For example, if P(x) is divided by 3x-2, let 3x+2=0, x=-2/3.  \newline P(-2/3) = Remainder (R(x))}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6J - Applications}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/liv-skreka_1711255419_Screenshot 2024-03-24 at 3.42.57 pm.jpeg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}