# 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})\$