Introduction To Polynomials
The word polynomial literally means many names. In maths this means an expression made by combining some constants, variables (also called $indeterminates$) using basic mathematical operations like addition, multiplication and exponentiation.
Let us take a look at the terms we used just now.
$\underline{Constant}$
As the name suggests this is a fixed value and does not change.
$\underline{Variable\ (or\ Indeterminate)}$
A symbol which does not have a fixed value. It can take any up any value. Each time the value changes the polynomial can take a new value.
Note: Polynomial are created using only non-negative, integer powers of its variables.
For example:
$3x^2y^3 + 17 x^2y - 23xy^2 - 7xy + 11$ is a polynomial in $x$ and $y$.
In general a polynomial is expressed as:
$a_nx^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \ldots \ + a_0$
where $a_n,\ a_{n-1},\ \ldots \ a_0$ are constant values (either negative or positive), and
each of $a_nx^n,\ a_{n-1}x^{n-1},\ a_{n-2}x^{n-2}\ldots$ is called a term.
-----------book page break-----------
$\underline{Degree\ of\ a\ term}$
The degree of any term in a polynomial is the sum of the powers of all the variables in the term.
For example the degree of the term $17x^2y$ our example above is $2 + 1 = 2$
The degree of the term $11$ is $0$, since we can write this term as $11x^0y^0$
$\underline{Degree\ of\ a\ polynomial}$
The degree of a polynomial is the degree of the term with the highest degree. In our example above the degree of the terms $3x^2y^3$ is $2 + 3 = 5$, which is the term with the highest degree. So, the degree of this polynomial is $5$.
$\underline{Like\ Terms}$
The terms in one or more polynomials that have the same power for each of the variables in the terms are called $Like\ Terms$
For example: $-11x^3y^4$ and $7x^3y^4$ are like terms, whereas,
the terms $3x^2y^3$ and $3x^3y^2$ are $not$ like terms, since the powers of individual variables are different.
Now let us try this problem:
--------- Reference to question: 736ddff0-d495-4bea-83e1-ba84d8b30d00 ---------
When two polynomials are added or subtracted, you can combine the like terms to form a single term. But you cannot combine unlike terms.
So, $3x^3y^2 - 14x^3y^2 = -11x^3y^2$ is correct. But you cannot combine two unlike terms like $7x^2y^3$ and $10x^3y^2$.
We will take an example of polynomial addition now.
-----------book page break-----------
We will find the value of $P(x) + G(x)$ where,
$P(x) = 7x^3y^3 - 11x^2y + 16xy^2 - 13xy + 19$
$G(x) = -5x^3y^3 + 17x^2y + 20xy - 14$
Here are the steps for adding these two polynomials:
$P(x) + G(x)$
$= (7x^3y^3 - 11x^2y + 16xy^2 - 13xy + 19) + (-5x^3y^3 + 17x^2y + 20xy - 14)$
$= 7x^3y^3 - 11x^2y + 16xy^2 - 13xy + 19 - 5x^3y^3 + 17x^2y + 20xy - 14$
$= \underline{7x^3y^3 - 5x^3y^3} \ \ \underline{- 11x^2y + 17x^2y} \ \ \underline{+ 16xy^2} \ \ \underline{- 13xy + 20xy} \ \ \underline{+ 19 - 14}$
$= \underline{2x^3y^3} \ \ \underline{+ 6x^2y} \ \ \underline{+ 16xy^2} \ \ \underline{+ 7xy} \ \ \underline{+ 5}$
$= 2x^3y^3 + 6x^2y + 16xy^2 + 7xy + 5$
III. Polynomial Subtraction
Now we will see an example of polynomial subtraction.
Find $P(x) - G(x)$ where,
$P(x) = 14x^4y^3 - 17x^3y^4 + x^3y^3 - 17x^2y^2 + 12x^2y + 7xy^2 + 15$
$G(x) = 17x^4y^3 - 11x^3y^4 - x^3y^3 - 15x^2y^2 + 12x^2y + 19xy - 12$
$= P(x) - G(x)$
$= (14x^4y^3 - 17x^3y^4 + x^3y^3 - 17x^2y^2 + 12x^2y + 7xy^2 + 15)$ $\ \ \ \ - (17x^4y^3 - 11x^3y^4 - x^3y^3 - 15x^2y^2 + 12x^2y + 19xy - 12)$
$= 14x^4y^3 - 17x^3y^4 + x^3y^3 - 17x^2y^2 + 12x^2y + 7xy^2 + 15$ $\ \ \ \ - 17x^4y^3 + 11x^3y^4 + x^3y^3 + 15x^2y^2 - 12x^2y - 19xy + 12$
(Notice above, that after opening the brackets the signs for all terms inside the bracket will change if the sign outside is negative)
-----------book page break-----------
$= \underline{14x^4y^3 - 17x^4y^3} \ \ \underline{- 17x^3y^4 + 11x^3y^4} \ \ \underline{+ x^3y^3 + x^3y^3} \ \ \underline{- 17x^2y^2 + 15x^2y^2}$ $\ \ \ \ \ \ \underline{+12x^2y - 12x^2y} \ \ \underline{+ 7xy^2} \ \ \underline{- 19xy} \ \ \underline{+ 12 + 15}$
$= \underline{- 3x^4y^3} \ \ \underline{- 6x^3y^4} \ \ \underline{+ 2x^3y^3} \ \ \underline{- 2x^2y^2} \ \ \underline{0} \ \ \underline{+ 7xy^2} \ \ \underline{- 19xy} \ \ \underline{+ 27}$
$= - 3x^4y^3 - 6x^3y^4 + 2x^3y^3 - 2x^2y^2 + 7xy^2 - 19xy + 27$
IV. Evaluating Polynomials
Given a polynomial $P(x) = 3x^2 + 11x - 22$, how much is $P(2)$?
This means that we need to find the value of the polynomial for the given at $x = 2$
$\texttip{\therefore}{therefore} p(2) = 3 \times 2^2 + 11 \times 2 - 22$
$= 3 \times 4 + 11 \times 2 - 22$
$= 12 + 22 - 22$
$= 12$
Like wise we can evaluate a polynomial with multiple variables. For example:
Find $P(3, 2)$ where $P(x, y) = x^2 + 12xy + 4y - 2y^2$
Substituting values $3, 2$ for variables $x, y$ respectively we get:
$P(3, 2) = 3^2 + 12 \times 3 \times 2 + 4 \times 2 - 2 \times 2^2$
$= 9 + 72 + 8 - 8$
$= 81$
-----------book page break-----------
Now let us try the following problem:
--------- Reference to question: be97bb4d-8cee-4852-9685-1cd65af3ee0a ---------