Posts for Tag: Algebra

Binomial Theorem – HSC Math

In elementary algebra, the binomial theorem or binomial expansion describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial $(x+y)^n$ into a sum involving terms of the form a $ax^by^c$, where the exponents $b$ and $c$ are nonnegative integers with $b+c=n$, and the coefficient $a$ ofRead More

Polynomial Equation – HSC Math

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate $x$ is $x^2−4x+7$. An example in three variables is $x^3+2xyz^2−yz+1$. Nature of roots of a quadraticRead More

Root Formulas, Algebra

Bases: a,b Powers (rational numbers): m,n $a,b≥0$ for even roots (n=2k, k∈N) $√^n{ab}=√^na.√^nb$ $√^na.√^mb=√^{nm}{a^mb^n}$ $√^n{a/b}={√^na}/{√^nb}$, b≠0 ${√^na}/{√^mb}={√^{nm}a^m}/{√^{nm}b^n}={√^{nm}{a^m/b^n}$, b≠0 $(√^na^m)^p=√^na^{mp}$ $√^na^n=a$ $√^na^m=√^{np}a^{mp}$ $√^na^m=a^{m/n}$ $√^m{√^na}=√^{mn}a$ $(√^na)^m=√^na^m$ $1/√^na={√^na^{n-1}}/a, a≠0$ $√{a±√b}=√{{a+√{a^2-b}}/2}±√{{a-√{a^2-b}}/2}$ $1/{√a±√b}={√a∓√b}/{a-b}$

Compound Interest Formulas

Future value: A Initial deposit: C Annual rate of interest: r Number of years invested: t Number of times compounded per year: n General Compound Interest Formula $A=C(1+r/n)^{nt}$ Simplified Compound Interest Formula If interest is compounded once per year, then the previous formula simplifies to: $A=C(1+r)^t$ Continuous Compound Interest If interest is compounded continually (n→∞),Read More

Inequalities

variables: x, y, z Real numbers: a, b, c, d …., m, n, … a1, 2, 3, … n Determinants: D, Dx, y, z Inequalities vs Interval notations: a ≤ x ≤ b      [a,b] a < x ≤ b      (a,b] a ≤ x < b      [a,b) a < x < b      (a,b) -∞ < xRead More

Equations and Solutions

Real numbers: a, b, c, p, q, u, v Solutions: x1, x2,y1, y2, y3 Linear equation with one variable $ax+b=0$ $x=-{b/a}$ Quadratic equation $ax^2+bx+c=0$ $x_{1,2}={-b±√{b^2-4ac}}/{2a}$ Discriminant $D=b^2-4ac$ Viete’s formulas if $x^2+px+q=0, \then$ $\{\table x_1+x_2=-p;x_1x_2=q$ if $ax^2+bx=0$, then $\{\table x_1=0;x_2=-b/a$ $ax^2+c=0$ $x_{1,2}=±√{-c/a}$ Cubic equation, Cardano’s formula. if $y^3+py+q=0$, then $\{\table y_1=u+v;y_{2,3}=-1/2(u+v)±√3/2(u+v)i$ where $u=√^3{-q/2+√{(q/2)^2+(p/3)^2}}$ and $v=√^3{-q/2-√{(q/2)^2-(p/3)^2}}$

Logarithm Formulas

Positive real numbers: x, y, a, c,k Natural numbers: n $y=log_ax$ if and only if $x=a^y, a>0, a≠1$ $log_a1=0$ $log_aa=1$ $log_a0=\{\table -∞ \if a>1;+∞ \if a0, c≠1$ $log_ac=1/{log_ca}$ $a^{log_ax}=x$ $log_10x=logx$, if base has not been mentioned it is base 10 $log_ex=ln x$, Natural logarithm. where $e={lim}↙{k→∞}(1+1/k)^k$ $logx=1/{ln10}lnx=0.434294$ ln x $ln x=1/{\log e}log x=2.302585$ $log x$

Power Formulas

Bases (positive real numbers): a,b Powers (rational numbers): m,n $a^ma^n=a^{m+n}$ $a^m/a^n=a^{m-n}=1/{a^{n-m}}$ $(ab)^m=a^mb^m$ $(a/b)^m={a^m}/{b^m}$ $(a^m)^n=a^{mn}$ $a^0=1, a≠0$ $a^1=1$ $a^{-1}=1/{a^m}$ $a^{m/n}=√^n{a^m}$

Product Formulas

Real numbers: a,b,c Whole numbers: n.k $(a+b)^2=a^2+2ab+b^2$ $(a-b)^2=a^2-2ab+b^2$ $(a+b)^3=a^3+3a^2b+3ab^2+b^3$ $(a-b)^3=a^3-3a^2b+3ab^2-b^3$ $(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$ $(a-b)^4=a^4-4a^3b+6a^2b^2-4ab^3+b^4$ $(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc$ $(a+b+c+…)^2=a^2+b^2+c^2+…+2(ab+ac+bc+…)$ Binomial Formula: $(a+b)^n=^nC_0a^n+^nC_1a^{n-1}b+^nC_2a^{n-2}b^2+…+^nC_{n-1}ab^{n-1}+^nC_nb^n$, where $^nC_k={n!}/{k!(n-k)!}$ are the binomial coefficients.

Factoring Formulas

Real numbers: a,b,c Natural number: n $a^2-b^2=(a-b)(a+b)$ $a^3-b^3=(a-b)(a^2+ab+b^2)$ $a^3+b^3=(a+b)(a^2-ab+b^2)$ $a^4-b^4=(a-b)(a+b)(a^2+b^2)$ $a^5-b^5=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)$ $a^5+b^5=(a-b)(a^4-a^3b+a^2b^2-ab^3+b^4)$ If n is odd, then $a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-…-ab^{n-2}+b^{n-1})$ If n is even, then $a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+…+ab^{n-2}+b^{n-1})$ $a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-…+ab^{n-2}-b^{n-1})$