Algebra - a MaS Tutor.

From these expansions we can see that the coefficients of each term are the elements of Pascal's Triangle, and the powers of $a$ count down to zero from left to right, whereas, the powers of $b$ count up from zero. So for the $n^{th}$ power ...

{(a + b)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) a^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a^{n - 1} b + (\begin{matrix} n \\ 2 \end{matrix}) a^{n - 2} b^{2} + \dots + (\begin{matrix} n \\ n - 2 \end{matrix}) a^{2} b^{n - 2} + (\begin{matrix} n \\ n - 1 \end{matrix}) a b^{n - 1} + (\begin{matrix} n \\ n \end{matrix}) b^{n}

where
$n \in ℕ$

This can be summed up as ...

{(a + b)}^{n} = Σ_{k = 0}^{n} (\begin{matrix} n \\ k \end{matrix}) a^{n - k} b^{k}

where
$k \in ℕ$
and
$(\begin{matrix} n \\ k \end{matrix}) = \frac{n!}{k! (n - k)!}$

As $n$ is a natural number then there are a finite number of terms in the expansion i.e., $(n + 1)$ terms.

A simpler expansion can be formed by letting $a = 1$ and $b = x$

{(1 + x)}^{n} = 1 + n x + \frac{n (n - 1) x^{2}}{2!} + \frac{n (n - 1) (n - 2) x^{3}}{3!} + ...

This again produces $(n + 1)$ terms if $n \in ℕ$ . However, this new form doesn't require $n$ to be a natural number. $n$ could be any real number, $n \in ℝ$ . If $n$ isn't a natural number then there are going to be an infinite number of terms because none of the factors in the righthand expression $n (n - 1) ... (n - r + 1)$ will equal to zero. The expansion is now not valid for all $x$ . It is only valid for $| x | < 1$ .

Expanding of ${(1 + a x)}^{n}$ , where n is negative or a fraction, is only valid for $| a x | < 1$ . This works in a similar way for more complex second terms.

Expanding ${(a + b x)}^{n}$ :

{(a + b x)}^{n} = {(a (1 + \frac{b x}{a}))}^{n} = a^{n} {(1 + \frac{b x}{a})}^{n}

The part ${(1 + \frac{b x}{a})}^{n}$ can then be expanded by one of the above methods.

The validity of the expansion is the same as before.

Worksheets
(Care of teacherworksheets.co.uk)

Binomial Expansion 1
(Care of www.physicsandmathstutor.com)

Binomial Expansion 2
(Care of www.physicsandmathstutor.com)

... search for more ...

Concatenation in Algebra

When mathematical symbols are concatenated; symbols are placed next to each other. In algebra, what is the significance of placing two or more mathematical symbols adjacent to one another within a term?

For A level and I/GCSE, there are at least three occasions where concatenation of symbols can occur.

As factors:

Examples,

2 x y

This means two times $x$ times $y$ .

3 p q^{3}

This means three times $p$ times $q$ cubed.

17 m (n + 2)

This means seventeen times $m$ times $n$ plus two.

As mixed numbers:

Examples,

2 \frac{2}{3}

This means two and two-thirds.

10 \frac{3}{5}

This means ten and three-fifths.

-2 \frac{1}{7}

This means minus two and one-seventh.

As functions:

Examples,

f (x)

This means the function, $f$ , possibly defined elsewhere, acts on $x$ .

sin x

This means the sine function acts on $x$ .

f g (x)

This means the function, $g$ acts on $x$ then $f$ acts on the result. This is known as a composite function.

Inequalities

a⁢x2 +b⁢x+c ≥0

a⁢x2 +b⁢x+c =0

y∝x

y∝1x

Proportion in Maths (Care of thirdspacelearning.com)

Proportion (Care of www.mathsgenie.co.uk)

Proportion (Care of revisionmaths.com)

... other Types of Porportion Square Root Proportionalityy∝x Square Proportionalityy∝x2 Inverse Square Proportionality y∝1x2

Square Root Proportionalityy∝x

Square Proportionalityy∝x2

Inverse Square Proportionality y∝1x2

Binomial Expansion

Prerequisite Materials:

Binomials

Expanding ( a+b⁢x ) n :

Worksheets (Care of teacherworksheets.co.uk)

Binomial Expansion 1 (Care of www.physicsandmathstutor.com)

Binomial Expansion 2 (Care of www.physicsandmathstutor.com)

Concatenation in Algebra

As factors: Examples,

As mixed numbers:Examples,

As functions:Examples,

$a x^{2} + b x + c \geq 0$

$a x^{2} + b x + c = 0$

$y \propto x$

$y \propto \frac{1}{x}$

Proportion in Maths
(Care of thirdspacelearning.com)

Proportion
(Care of www.mathsgenie.co.uk)

Proportion
(Care of revisionmaths.com)

... other Types of Porportion

Square Root Proportionality $y \propto \sqrt{x}$

Square Proportionality $y \propto x^{2}$

Inverse Square Proportionality $y \propto \frac{1}{x^{2}}$

Square Root Proportionality $y \propto \sqrt{x}$

Square Proportionality $y \propto x^{2}$

Inverse Square Proportionality $y \propto \frac{1}{x^{2}}$