UPSKILL MATH PLUS

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### Theory:

The general form of polynomial expression in one variable($$x$$):

For the variable $x$, a polynomial $p\left(x\right)$ is an algebraic expression in the form of $x$ can be written as follows.

$$p(x) =$$ $$a_{n}x^{n}$$$$a_{n-1}x^{n-1}$$$$+...$$$$a_{2}x^{2}$$$$+$$$$a_{1}x$$$$+a_{0}$$

Where ${a}_{0}$, ${a}_{1}$, ${a}_{2}$, ......, ${a}_{n}$ are constants, ${a}_{n}\ne 0$ and $$n$$ is a non-negative integer.

Important!
The exponent of the variable in the polynomial is the non-negative integer(whole number).
• The area of the square can be written as a polynomial in the following form.
$p\left(x\right)\phantom{\rule{0.147em}{0ex}}={x}^{2}$.     (Where the variable is '$$x$$')
•  The perimeter of the square can be written as a polynomial in the following form.
$q\left(x\right)\phantom{\rule{0.147em}{0ex}}=4x$.      (Where the variable is '$$x$$')
• The area of a circle can be written as a polynomial in the following form.
$q\left(r\right)\phantom{\rule{0.147em}{0ex}}=\mathrm{\pi }{r}^{2}$.     (Where the variable is '$$r$$')
• The perimeter of the circle can be written as a polynomial in the following form.
$q\left(r\right)\phantom{\rule{0.147em}{0ex}}=4\mathrm{\pi }r$.         (Where the variable is '$$r$$')
• Consider a polynomial with degree '$$3$$'.
${x}^{3}+{x}^{2}+6x+9$.     (Where the variable is '$$x$$')

Such polynomials are referred to as polynomials in one variable.

Important!
Note that all the above polynomials have only one variable.
Let us see some examples on polynomials of more than one variable.
• The area of a rectangle can be written as a polynomial in the following form.
$p\left(x,y\right)\phantom{\rule{0.147em}{0ex}}=\mathit{xy}$.    (Where the variables are '$$x$$', and '$$y$$')
• The perimeter of the rectangle can be represented as a polynomial in the following form.
$q\left(x,y\right)\phantom{\rule{0.147em}{0ex}}=2x+2y$.     (Where the variables are '$$x$$', and '$$y$$')
• The area of a cylinder is represented as a polynomial in the following form.
$q\left(r,h\right)\phantom{\rule{0.147em}{0ex}}=\mathrm{\pi }{r}^{2}h$.       (Where the variables are '$$r$$', and '$$h$$')
• The perimeter of the cylinder can be represented as a polynomial in the following form.
$q\left(r,h\right)\phantom{\rule{0.147em}{0ex}}=4\mathrm{\pi }\mathit{rh}$.        (Where the variables are '$$r$$', and '$$h$$')

Important!
Note that all four above polynomials have two variables.
• The area of a trapezium can be represented as the following polynomial.
$p\left(x,y,h\right)=\frac{1}{2}\left(x+y\right)×h$   (Where the variables are '$$x$$', '$$y$$', and '$$h$$')
• Consider the algebraic expression ${x}^{3}+{y}^{2}+z+6\mathit{xyz}+9$.  (Where the variables are '$$x$$', '$$y$$', and '$$z$$')

Important!
Note that the above polynomials have three variables.
• Consider the algebraic expression with four variables.
${x}^{3}+{y}^{2}+z+2p+6\mathit{xyz}+9$  (Where variables are '$$x$$', '$$y$$', '$$z$$', and '$$p$$')

Important!
Note that the above polynomials have four variables.