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APEX Calculus: for University of Lethbridge

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*}