Skip to main content Contents Index Calc Prev Up Next \(\require{cancel}\newcommand{\highlight}[1]{{\color{blue}{#1}}}
\newcommand{\ds}{\displaystyle}
\newcommand{\fp}{f'}
\newcommand{\fpp}{f''}
\newcommand{\lz}[2]{\frac{d#1}{d#2}}
\newcommand{\lzn}[3]{\frac{d^{#1}#2}{d#3^{#1}}}
\newcommand{\lzo}[1]{\frac{d}{d#1}}
\newcommand{\lzoo}[2]{{\frac{d}{d#1}}{\left(#2\right)}}
\newcommand{\lzon}[2]{\frac{d^{#1}}{d#2^{#1}}}
\newcommand{\lzoa}[3]{\left.{\frac{d#1}{d#2}}\right|_{#3}}
\newcommand{\plz}[2]{\frac{\partial#1}{\partial#2}}
\newcommand{\plzoa}[3]{\left.{\frac{\partial#1}{\partial#2}}\right|_{#3}}
\newcommand{\inflim}[1][n]{\lim\limits_{#1 \to \infty}}
\newcommand{\infser}[1][1]{\sum_{n=#1}^\infty}
\newcommand{\Fp}{F\primeskip'}
\newcommand{\Fpp}{F\primeskip''}
\newcommand{\yp}{y\primeskip'}
\newcommand{\gp}{g\primeskip'}
\newcommand{\dx}{\Delta x}
\newcommand{\dy}{\Delta y}
\newcommand{\ddz}{\Delta z}
\newcommand{\thet}{\theta}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\newcommand{\vnorm}[1]{\left\lVert\vec #1\right\rVert}
\newcommand{\snorm}[1]{\left|\left|\,#1\,\right|\right|}
\newcommand{\la}{\left\langle}
\newcommand{\ra}{\right\rangle}
\newcommand{\dotp}[2]{\vec #1 \cdot \vec #2}
\newcommand{\proj}[2]{\text{proj}_{\,\vec #2}{\,\vec #1}}
\newcommand{\crossp}[2]{\vec #1 \times \vec #2}
\newcommand{\veci}{\vec i}
\newcommand{\vecj}{\vec j}
\newcommand{\veck}{\vec k}
\newcommand{\vecu}{\vec u}
\newcommand{\vecv}{\vec v}
\newcommand{\vecw}{\vec w}
\newcommand{\vecx}{\vec x}
\newcommand{\vecy}{\vec y}
\newcommand{\vrp}{\vec r\hskip0.75pt '}
\newcommand{\vrpp}{\vec r\hskip0.75pt ''}
\newcommand{\vsp}{\vec s\hskip0.75pt '}
\newcommand{\vrt}{\vec r(t)}
\newcommand{\vst}{\vec s(t)}
\newcommand{\vvt}{\vec v(t)}
\newcommand{\vat}{\vec a(t)}
\newcommand{\px}{\partial x}
\newcommand{\py}{\partial y}
\newcommand{\pz}{\partial z}
\newcommand{\pf}{\partial f}
\newcommand{\unittangent}{\vec{{}T}}
\newcommand{\unitnormal}{\vec{N}}
\newcommand{\unittangentprime}{\vec{{}T}\hskip0.75pt '}
\newcommand{\R}{mathbb{R}}
\newcommand{\mathN}{\mathbb{N}}
\newcommand{\surfaceS}{\mathcal{S}}
\newcommand{\zerooverzero}{\ds \raisebox{8pt}{\text{``\ }}\frac{0}{0}\raisebox{8pt}{\textit{ ''}}}
\newcommand{\deriv}[2]{\myds\frac{d}{dx}\left(#1\right)=#2}
\newcommand{\myint}[2]{\myds\int #1\, dx= {\ds #2}}
\newcommand{\primeskip}{\hskip.75pt}
\newcommand{\abs}[1]{\left\lvert #1\right\rvert}
\newcommand{\sech}{\operatorname{sech}}
\newcommand{\csch}{\operatorname{csch}}
\newcommand{\curl}{\operatorname{curl}}
\newcommand{\divv}{\operatorname{div}}
\newcommand{\Hess}{\operatorname{Hess}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section B.5 Algebra
Factors and Zeros of Polynomials. Let \(p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\) be a polynomial. If \(p(a)=0\text{,}\) then \(a\) is a \(zero\) of the polynomial and a solution of the equation \(p(x)=0\text{.}\) Furthermore, \((x-a)\) is a \(factor\) of the polynomial.
Fundamental Theorem of Algebra. An \(n\) th degree polynomial has \(n\) (not necessarily distinct) zeros. Although all of these zeros may be imaginary, a real polynomial of odd degree must have at least one real zero.
Quadratic Formula. If \(p(x) = ax^2 + bx + c\text{,}\) and \(0 \le b^2 - 4ac\text{,}\) then the real zeros of \(p\) are \(x=(-b\pm \sqrt{b^2-4ac})/2a\)
Special Factors.
\begin{align*}
x^2 - a^2 \amp = (x-a)(x+a)\\
x^3 - a^3 \amp= (x-a)(x^2+ax+a^2)\\
x^3 + a^3 \amp= (x+a)(x^2-ax+a^2)\\
x^4 - a^4 \amp= (x^2-a^2)(x^2+a^2)\\
(x+y)^n \amp=x^n + nx^{n-1}y+\frac{n(n-1)}{2!}x^{n-2}y^2+\cdots +nxy^{n-1}+y^n\\
(x-y)^n \amp=x^n - nx^{n-1}y+\frac{n(n-1)}{2!}x^{n-2}y^2-\cdots \pm nxy^{n-1}\mp y^n
\end{align*}
Binomial Theorem.
\begin{align*}
(x+y)^2 \amp= x^2 + 2xy + y^2\\
(x-y)^2 \amp= x^2 -2xy +y^2\\
(x+y)^3 \amp= x^3 + 3x^2y + 3xy^2 + y^3\\
(x-y)^3 \amp= x^3 -3x^2y + 3xy^2 -y^3\\
(x+y)^4 \amp= x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\\
(x-y)^4 \amp= x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4
\end{align*}
Rational Zero Theorem. If \(p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\) has integer coefficients, then every \(rational\) \(zero\) of \(p\) is of the form \(x=r/s\text{,}\) where \(r\) is a factor of \(a_0\) and \(s\) is a factor of \(a_n\text{.}\)
Factoring by Grouping. \(ac x^3 + adx^2 + bcx + bd = ax^2(cx+d)+b(cx+d)=(ax^2+b)(cx+d)\)
Arithmetic Operations.
\begin{align*}
ab+ac\amp=a(b+c) \amp \frac{a}{b}+\frac{c}{d} \amp= \frac{ad+bc}{bd} \amp \frac{a+b}{c} \amp = \frac{a}{c} + \frac{b}{c}\\
\frac{\left(\displaystyle\frac{a}{b}\right)}{\left(\displaystyle\frac{c}{d}\right)}\amp=\left(\frac{a}{b}\right)\left(\frac{d}{c}\right)=\frac{ad}{bc} \amp \frac{\left(\displaystyle\frac{a}{b}\right)}{c} \amp = \frac{a}{bc} \amp \frac{a}{\left(\displaystyle\frac{b}{c}\right)} \amp= \frac{ac}{b}\\
a\left(\frac{b}{c}\right)\amp= \frac{ab}{c}\amp \frac{a-b}{c-d}\amp=\frac{b-a}{d-c}\amp \frac{ab+ac}{a}\amp=b+c
\end{align*}
Exponents and Radicals.
\begin{align*}
a^0\amp =1, \, a \ne 0 \amp (ab)^x\amp=a^xb^x \amp a^xa^y \amp = a^{x+y} \amp \sqrt{a}\amp=a^{1/2}\\
\frac{a^x}{a^y}\amp=a^{x-y} \amp \sqrt[n]{a}\amp =a^{1/n} \amp \left(\frac{a}{b}\right)^x\amp=\frac{a^x}{b^x} \amp \sqrt[n]{a^m}\amp=a^{m/n}\\
a^{-x}\amp=\displaystyle\frac{1}{a^x} \amp \sqrt[n]{ab}\amp=\sqrt[n]{a}\sqrt[n]{b} \amp (a^x)^y\amp=a^{xy} \amp \sqrt[n]{\frac{a}{b}}\amp=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}
\end{align*}
