Real Nos.

Different Types of Real Nos.

• A number that can be represented as a point on the number line is a Real No.
• Such number line is called Real No. Line.
• There are several types of numbers within the Real No. Line.
• In Elementary/ Primary schools, understanding of Natural nos. and some knowledge on fractions is good enough for Math study.
• In middle/ high schools, Directed nos., Rational and Irrational nos. are introduced.

Definitions

1. Natural nos. : $\{1, 2, 3, \ldots \} \$ denoted by the set $\mathbb{N} \$ [some authors include integer 0]
2. Integers : $\{\ldots , -2, -1, 0, 1, 2, \ldots \} \$ denoted by the set $\mathbb{Z}$
3. Rational nos. : $\big\{ \frac{p}{q}\big |\; p, q \in \mathbb{Z},\; q \neq 0 \big\} \$ denoted by the set $\mathbb{Q}$
4. Irrational nos. : $\mathbb{R} {\setminus} \mathbb{Q}$
5. Surds $\subseteq \mathbb{R} {\setminus} \mathbb{Q}$. Examples are: square root $(\sqrt{\phantom{x}})$, cube root $(\sqrt[3]{\phantom{x}})$ form, and so on.

Misconceptions

1. Fractions and Rational nos. seems to be the same but they are different. All fractions are Rational nos., not all rational nos. are fractions. By definition, all integers are Rational nos., but integers are commonly considered different from fractions.
2. All decimal nos. can be expressed in the form of Rational nos. no matter recurring or finite decimal. Thus, the set of decimal nos. is not a separate type of Real nos., it is a subset of Rational nos.

Problem Solving Techniques

I.   Convert a recurring decimal no. to fraction, e.g. $$0.23\dot 45\dot 6$$

1. Decompose non-recurring and recurring part of the decimal
2. Construct a fraction with denominator having 9 for each recurring digit and 0 for each non-recurring digit. 有循環就9, 無循環就0, 先9後0, and with recurring digits as numerator.

For this example,

Polynomial

Definitions

• Either a number or a number (called coefficient) multiplied by a variable is called a/an (algebraic) term
• 2 terms with the same variables raised to the same power, no matter if the numbers are the same are called like terms
• Algebraic expression is an expression made up of adding multiple algebraic terms.
• Polynomial is a kind of algebraic expression with a restriction that the power/exponent/index of the variable(s) must be non-negative integers
• Polynomial with one, two and three terms are called monomials, binomials, and trinomials respectively.
• The degree of a polynomial is the maximum value of the sums of the exponents of the variables in each term
• If a polynomial can be expressed as a product of polynomials, then the process of factoring this polynomial to the maximum no. of factors is called factorization

Useful Identities for Factorization

Identities are special kind of equations such that the left-hand-side (L.H.S.) is equal to the right-hand-side (R.H.S.) for all possible values of the variables, and the equation sign may be denoted by the equivalent sign $\equiv .$

1. $x^2 – y^2 \equiv (x+y)(x-y)$
2. $x^2 + 2xy + y^2 \equiv (x+y)^2$
3. $x^2-2xy + y^2 \equiv (x-y)^2$
4. $x^3 + y^3 \equiv (x+y)(x^2-xy + y^2)$
5. $x^3-y^3 \equiv (x-y)(x^2+xy+y^2)$

Example

Factorize:

$\mbox{ a) } \quad a^2- b^2-2a +1 \\ \mbox{ b) } \quad 3x^3- 24 \\[.5em]$

Solution:

\begin{align} \mbox{ a) } \quad & a^2-b^2-2a+1 \\ = \, & (a^2-2a+1)-b^2 \\ = \, & (a-1)^2-b^2 \\ = \, &(a-1+b)(a-1-b) \\[.5em] \mbox{ b) } \quad & 3x^3-24 \\ = \, & 3(x^3-8) \\ = \, & 3(x^3-2^3) \\ = \, & 3(x-2)(x^2+2x+4) \\ \end{align}

Remainder & Factor Theorems

The Remainder Thm. states that
if a polynomial $f(x)$ is divided by $x-a$, then the remainder
$\quad R = f(a).$

Consider the divisor $(x-a)$ of the polynomial $f(x)$ with quotient $Q(x)$ and the remainder $R, \$ we have:

\quad \begin{align} f(x) &= (x-a)Q(x) + R, \\[.3em] \mbox{ put } x &= a, \mbox{ we have } \\[.3em] f(a) &= (a-a)Q(a) + R \\ &= R \end{align} \\[2em]

Example

When ${kx}^2 + x -1 \,$ is divided by $x+2, \,$ the remainder is 1. Find $k.$

Solution:

\mbox{Let } \, f(x) = kx^2+x-1 \\ \begin{align} \because \kern4.5em f(-2) &= 1 \\ \therefore k(-2)^2+(-2)-1 &= 1 \\ 4k-3 &= 1 \\ k &= 1 \end{align}

The Factor Thm. states that
a polynomial $f(x)$ has a factor $\, (x-k) \,$ if and only if $\, f(k) = 0$

Example

Let $\, f(x) = 4x^3-8x^2-3x+9$

\begin{align} \mbox{a) } & \mbox{ Show that } \, x+1 \, \mbox{ is a factor of } \, f(x) \\[.5em] \mbox{b) } & \mbox{ Factorize } f(x) \\[.5em] \mbox{c) } & \mbox{ Given that } g(x) = x(x+1)^2(2x-3)^3 \\ & \mbox{ Find the H.C.F. and L.C.M. of } \\ & \quad f(x) \mbox{ and } g(x). \\ \end{align}

Solution:

\begin{align} \mbox{a)} \quad f(-1) &= 4(-1)^3-8(-1)^2-3(-1) + 9 \\ &= {-4}-8+3+9 \\ &=0 \end{align} \\ \qquad \therefore x+1 \mbox{ is a factor of } f(x) \\[1.5em]
$\require{enclose} \begin{array}{lr} \mbox{b) } & \kern-4ex 4x^2-12x+9 \\[-3pt] & x+1 \enclose{longdiv}{4x^3- \phantom{0}8x^2- \phantom{0}3x+9} \\[-2ex] & \underline{4x^3+ \phantom{0} 4x^2} \kern9ex \\ & -12x^2- \phantom{0} 3x \kern3.5ex \\[-2ex] & \underline{ -12x^2-12x} \kern3.5ex \\[-3pt] & 9x+9 \\[-2ex] & \underline{9x+9} \\[.6em] & \therefore f(x) = (x+1)(4x^2-12x+9) \\ & = (x+1)(2x-3)^2 \kern5.5ex \\ \end{array}$

$\begin{array}{lll} \mbox{c) } & \mbox{H.C.F.} &\kern-2ex = (x+1)(2x-3)^2 \\ & \mbox{L.C.M.} &\kern-2ex = x(x+1)^2(2x-3)^3 \\ \end{array}$