Subsection A.3 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.
\(\begin{array}{ll} x^2 - a^2 = (x-a)(x+a) & x^3 - a^3 = (x-a)(x^2+ax+a^2)\\ x^3 + a^3 = (x+a)(x^2-ax+a^2) \quad & x^4 - a^4 = (x^2-a^2)(x^2+a^2)\\ \end{array} \)
\(\begin{array}{l} (x+y)^n =x^n + nx^{n-1}y+\frac{n(n-1)}{2!}x^{n-2}y^2+\cdots +nxy^{n-1}+y^n\\ (x-y)^n =x^n - nx^{n-1}y+\frac{n(n-1)}{2!}x^{n-2}y^2-\cdots \pm nxy^{n-1}\mp y^n \end{array}\)
Binomial Theorem.
\(\begin{array}{l} (x+y)^2 = x^2 + 2xy + y^2 \\ (x-y)^2 = x^2 -2xy +y^2 \\ \\ (x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 \qquad \\ (x-y)^3 = x^3 -3x^2y + 3xy^2 -y^3\\ \\ (x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4 \qquad \\ (x-y)^4 = x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4\\ \end{array} \)
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 \(a_n\text{.}\)
Arithmetic Operations.
\(\begin{array}{ll} ab+ac=a(b+c) \qquad \qquad & \displaystyle \frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{bd} \\ \\ \displaystyle \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c} & \displaystyle \frac{\left(\displaystyle \frac{a}{b}\right)}{\left(\displaystyle \frac{c}{d}\right)}=\left(\frac{a}{b}\right)\left(\frac{d}{c}\right)=\frac{ad}{bc} \\ \\ \displaystyle \frac{\left(\displaystyle \frac{a}{b}\right)}{c} =\displaystyle \frac{a}{bc} &\displaystyle \frac{a}{\left(\displaystyle \frac{b}{c}\right)} =\displaystyle \frac{ac}{b} \\ \\ a\left(\displaystyle \frac{b}{c}\right)= \displaystyle \frac{ab}{c} &\displaystyle \frac{a-b}{c-d}=\frac{b-a}{d-c} \\ \\ \displaystyle \frac{ab+ac}{a}=b+c\\ \end{array} \)
Exponents and Radicals.
\(\begin{array}{lll} a^0=1, \; a \ne 0 \quad & (ab)^x=a^xb^x \quad & a^xa^y = a^{x+y} \\ \\ \sqrt{a}=a^{1/2} &\displaystyle \frac{a^x}{a^y}=a^{x-y} & \sqrt[n]{a}=a^{1/n} \\ \\ \left(\displaystyle \frac{a}{b}\right)^x=\displaystyle \frac{a^x}{b^x} &\sqrt[n]{a^m}=a^{m/n} & a^{-x}=\displaystyle \frac{1}{a^x} \\ \\ \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b} & (a^x)^y=a^{xy} & \displaystyle \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}} \end{array} \)