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algebra

Exponents & logs, sequences & series, counting & binomial, polynomials, partial fractions, proof, linear systems.

exponents & logs · equations & inequalities · sequences & series · counting & binomial · polynomials & partial fractions · proof & linear systems

Exponents & logs

Exponents. \((ab)^n=a^nb^n\), \(a^ma^n=a^{m+n}\), \(a^m/a^n=a^{m-n}\), \((a^m)^n=a^{mn}\), \(a^{-n}=1/a^n\), \(a^{p/q}=\sqrt[q]{a^p}\). \(a^x=e^{x\ln a}\). \(\sqrt{ab}=\sqrt a\sqrt b\) only safely for \(a,b\ge0\).

Logarithms. \(a^x=b\Leftrightarrow x=\log_a b\). \(\log_a xy=\log_a x+\log_a y\), \(\log_a(x/y)=\log_a x-\log_a y\), \(\log_a x^r=r\log_a x\); change of base \(\log_a x=\dfrac{\log_b x}{\log_b a}=\dfrac{\ln x}{\ln a}\).

Equations & inequalities

Linear & quadratic. \(ax+b=0\Rightarrow x=-b/a\). \(ax^2+bx+c=0\Rightarrow x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\). Discriminant \(\Delta=b^2-4ac\): \(\Delta\gt0\) two real roots, \(\Delta=0\) repeated, \(\Delta\lt0\) no real roots. Completed square: \(ax^2+bx+c=a\left(x+\dfrac b{2a}\right)^2+c-\dfrac{b^2}{4a}\).

Inequalities & modulus. \(|u|=\sqrt{u^2}\). \(|u|\le k\Leftrightarrow -k\le u\le k\) \((k\ge0)\). \(|u|\ge k\Leftrightarrow u\le-k\) or \(u\ge k\). \(|u|=k\Leftrightarrow u=\pm k\). \(|u|\lt|v|\Leftrightarrow u^2\lt v^2\). \(|x-a|\lt r\Leftrightarrow a-r\lt x\lt a+r\); \(|x-a|\gt r\Leftrightarrow x\lt a-r\) or \(x\gt a+r\).

Sign charts. Zeros and poles partition the line into intervals; even multiplicity gives no sign change, odd multiplicity flips the sign.

Sequences & series

Arithmetic. \(u_n=u_1+(n-1)d\); \(S_n=\dfrac n2\bigl(2u_1+(n-1)d\bigr)=\dfrac n2(u_1+u_n)\).

Geometric. \(u_n=u_1r^{n-1}\); \(S_n=\dfrac{u_1(r^n-1)}{r-1}=\dfrac{u_1(1-r^n)}{1-r}\) \((r\ne1)\); \(S_\infty=\dfrac{u_1}{1-r}\) for \(|r|\lt1\).

Standard sums. \(\sum_{r=1}^{n}r=\dfrac{n(n+1)}2\), \(\sum r^2=\dfrac{n(n+1)(2n+1)}6\), \(\sum r^3=\left(\dfrac{n(n+1)}2\right)^2\). Telescoping: \(\sum(a_r-a_{r+1})=a_1-a_{n+1}\).

Finance & growth. Simple interest \(FV=PV(1+in)\); compound \(FV=PV\left(1+\dfrac{r}{100k}\right)^{kn}\); depreciation \(V=V_0(1-d/100)^n\). Real growth factor \(\dfrac{1+i}{1+j}\); real percentage \(=\dfrac{1+i}{1+j}-1\). Continuous model \(N=N_0e^{kt}\): doubling time \(t=\ln2/k\), half-life \(t=\ln(1/2)/k\).

Counting & binomial

Counting. Product rule \(N=n_1n_2\cdots n_k\). \({}^nP_r=\dfrac{n!}{(n-r)!}\), \({}^nC_r=\dbinom nr=\dfrac{n!}{r!\,(n-r)!}\); \(\dbinom nr=\dbinom n{n-r}\), \(\dbinom nr+\dbinom n{r+1}=\dbinom{n+1}{r+1}\).

Binomial theorem. \((a+b)^n=\sum_{r=0}^{n}\dbinom nr a^{n-r}b^r\); general term \(T_{r+1}=\dbinom nr a^{n-r}b^r\).

Generalized binomial. \((1+x)^p=1+px+\dfrac{p(p-1)}{2!}x^2+\dfrac{p(p-1)(p-2)}{3!}x^3+\cdots\) for \(|x|\lt1\), \(p\in\Q\). \((a+b)^p=a^p(1+b/a)^p\), needs \(|b/a|\lt1\).

Polynomials & partial fractions

Polynomials. \(P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0\). Remainder theorem: the remainder on division by \((x-a)\) is \(P(a)\). Factor theorem: \((x-a)\mid P(x)\Leftrightarrow P(a)=0\). With roots \(\alpha_i\): \(P(x)=a_n\prod_{i=1}^n(x-\alpha_i)\); Vieta: \(\sum\alpha_i=-a_{n-1}/a_n\), \(\prod\alpha_i=(-1)^na_0/a_n\). Real coefficients: non-real roots occur in conjugate pairs. Multiplicity \(m\): graph crosses if \(m\) odd, touches if \(m\) even. Division: \(P=DQ+R\) with \(\deg R\lt\deg D\).

Partial fractions. Requires numerator degree \(\lt\) denominator degree. \(\dfrac{px+q}{(x-a)(x-b)}=\dfrac{A}{x-a}+\dfrac{B}{x-b}\) with \(A=\dfrac{pa+q}{a-b}\), \(B=\dfrac{pb+q}{b-a}\). Repeated and irreducible-quadratic patterns: \(\dfrac{P(x)}{(x-a)^2}=\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}\); \(\dfrac{P(x)}{(x-a)(x^2+px+q)}=\dfrac{A}{x-a}+\dfrac{Bx+C}{x^2+px+q}\).

Proof & linear systems

Proof. Induction: prove \(P(n_0)\), assume \(P(k)\), prove \(P(k+1)\). Contradiction: assume the negation, derive an impossibility. Counterexample: one valid case falsifies a universal statement.

Linear systems. Augmented matrix \([A\mid b]\): unique solution iff \(\rank A=\rank[A\mid b]=n\); infinitely many iff \(\rank A=\rank[A\mid b]\lt n\); none iff \(\rank A\lt\rank[A\mid b]\). RREF pivots give the leading variables; non-pivot columns are parameters.