Polynomials represent an unknown quantity by a variable. Usually, alphabets and some other symbols are used as variables. A variable can take any real value. For example, x, y, z are variables.
An Algebraic expression is made up of terms. A term may be a single variable or a constant or a product of a variable and a number. In general, an algebraic expression is of the form (constant) × (variable) $\frac{1}{2}$ x, x, 7x are examples of algebraic expressions. $\frac{1}{2}$ x - The variable x is multiplied by constant $\frac{1}{2}$

x - The variable x is multiplied by constant 1.
7x - the variable x is multiplied by constant 7.

If we do not know the value of the constant, then we usually use the letters a, b, c, d to represent the constant. When we say ‘ay', ‘a' represents a constant unless mentioned otherwise and ‘y' represents a variable. The value of a constant does not change but the value of a variable can change in a problem. Each term in an algebraic expression is separated by a + or - sign. A monomial is an algebraic expression with a single algebraic term. The Numerical Coefficient of a monomial is the numerical part of the monomial. For example, the velocity of an object dropped from a height of s units can be represented as v2 = 2gs where g is acceleration due to gravity. In this algebraic equation, v$^{2}$ and 2gs are terms. They are monomials. 2 is the Numerical Coefficient of the term 2gs. Every term in a polynomial will have a coefficient. Consider the polynomial -x + 3. This polynomial can be rewritten as -x + 3x$^{0}$ The coefficient of x is -1. The coefficient of x$^{0}$ is 3 or 3 is the coefficient of x$^{0}$

An algebraic expression which consists of two or more terms, is called a polynomial
Example: 5x-2, 3x+7y

Term:
A term is a constant, a variable, or a product or quotient of numbers and variables.

Example:
(i) 7 is a term, which is a constant
(ii) y is a term, which is a variable
(iii) 9x is a term, which is a product of a constant and a
Variable
(iv) $\frac{7}{z}$ is a term, which is a quotient of a constant and
a variable

Like Terms:

The terms having the same literal coefficients are called like terms. The like terms may differ only in their numeral coefficients.

Example: 2xy, xy, -9xy are like terms
(each having same literal coefficient xy)

Unlike Terms:

The terms which do not have same literal coefficients are called unlike terms.

Example: 2a, 3ab, 6ac are unlike terms

Coefficient of a term:

Any factor or a group of factors of a product are known as the coefficient of remaining factors.

Example:

In product 7xy, 7 is numeral coefficient of xy
7x is the coefficient of y
xy is the coefficient of 7 and so on!


When the coefficient is unity, that is 1 (one), it is usually omitted, 1x is written as x.
A polynomial is a mathematical expression of the form
P(x) = an xn + an-1 xn-1 +...........+a1 x1 + a0 where an $\neq$ 0
an, an-1, ........a1, a0 are called constants.
A polynomial with one term is called a monomial. Polynomials with two terms are Binomials. Similarly, a polynomial with three terms is called a Trinomial.

There are 3 types of polynomials

Monomials : A monomial is an algebraic expression with a single algebraic term. The Numerical Coefficient of a monomial is the numerical part of the monomial.

4x$^{3}$ is a Monomial

Bionomial : A Binomial is a polynomial with exactly two terms

4x$^{3}$ - 3 is a Bionomial

Trinomial : a Trinomial is a polynomial with exactly three terms

x + x$^{3}$ + x$^{2}$ is a Trinomial.

When we are adding polynomials, we might encounter 4 different scenarios. Let's look at them one by one.

1. When all the terms are positive, add their coefficients.

Example: Add 3xy, 5xy and 8xy
Solution:
The sum of the coefficients= 3+5+8
= 16

So, 3xy +5xy+8xy = 16 xy Answer

Alternate method:
3xy +5xy + 8xy = (3+5+8). xy
= 16 xy Answer

2. When all the terms are negative, add their coefficients without considering their negative signs and then prefix the minus sign to the sum.

Example: Add -3xy, -5xy and -xy

Solution: Without considering the negative signs, the coefficients of the given terms are 3, 5 and 1 respectively and 3+5+1 = 9

Addition of -3xy, -5xy and -xy = -9xy

(-3xy) + (-5xy) + (-xy) = -9xy Answer

Alternate Method:

(-3xy) + (-5xy) + (-xy)
Taking out the negative sign, we get

= - ( 3xy +5xy+xy)
= - [(3+5+1)xy]
= - (9) xy
= - 9xy Answer

3. When all the terms are not of the same sign . The same rule as the one used for the addition of integers should be applied.

Example: Addition of 17x and -5x
Solution: 17x + (-5x)
= 17x -5x
= (17-5) . x
= 12 x Answer

Example : Simplify (3x+4y) + (7x-2y
Solution :
We can solve this problem using any one of the two methods

1. (Adding horizontally)

First we need to open the parenthesis and then simplify the
like terms,
(3x+4y) + (7x-2y)
= 3x +4y +7x -2y (simplify the like terms in x and y)
= 3x +7x +4y-2y
= (3+7). x + (4-2). y
= 10x +2y
So, (3x+4y) + (7x-2y) = 10x + 2y Answer

2.Alternate Method : (Adding vertically)
Steps to be followed:

1. Arrange the terms of each of the given polynomials
either alphabetically or in descending powers of some
variable

2. Arrange the given polynomials in the form of rows in
such a way that the like terms occur in the same
column

3. Combine the like terms column wise.

Example: (3x+4y) + (7x-2y)

3x + 4y
(+) 7x - 2y
10x + 2y ( 7+3 = 10 and 4-2 = 2)

So, (3x+4y) + (7x-2y) = 10x + 2y Answer

(Thus, we can solve the given problem using any one of the two given methods)

4. Addition of unlike terms: The sum of two or more like terms is a single like term ; but the two unlike terms cannot be added together to get a single term

Example:

The unlike terms 3ab and 5bc cannot be added together to form a single term. All that which can be done is we can connect them by the sign of addition and write the result in the form:
Addition of 3ab and 5bc is 3ab + 5bc Answer

When we are subtracting polynomials, we might encounter 4 different scenarios. Let's look at them one by one.

Subtraction of like terms:
The same rule, as those used for the subtraction of integers are applied for the subtraction of like terms. In simple words, when two like terms have different signs, we need to subtract them and give them the sign of the term with the greater coefficient.

Example: subtract 5x from 2x
Solution:
2x - 5x
= (2-5). x (2 - 5 = -3)
= -3x Answer
Note: The result of the subtraction of two like terms is also
a like term!

Subtraction of unlike terms:
We cannot get a single term by the subtraction of unlike terms.

Example: 2ab, 4bc are two unlike terms
The subtraction of 2ab from 4bc is 4bc - 2ab Answer

Example: Evaluate 7ax + ax - 4ax
Solution:

Add the positive terms together and the negative terms together separately. Then we need to find out the result of the two terms obtained on simplification.

7ax + ax - 4ax
= (7+1).ax - 4ax (7ax and ax are positive terms)
= 8 ax - 4 ax
= (8-4). ax
= 4ax Answer

Subtracting using the vertical method:
Step1: Arrange the terms of the given polynomial in the same order
Step2: Write the given expressions in two rows in such a way that the like terms are written one below the other, keeping the expression to be subtracted in the second row
Step3: Change the sign of each term in the lower row from + to - and - to +
Step4: With new signs of the terms of the lower row, add column wise.

Example:
Subtract 3a + 4b - 2c from 6a - 2b + 3c

Solution: We have,
6a - 2b + 3c
(-) + 3a + 4b - 2c
- - +
3a - 6b + 5c

So, (6a - 2b +3c) - (3a + 4b - 2c) = 3a - 6b + 5c Answer

Subtraction Horizontally:
Step1: Write the two expressions in one row with the expression to be subtracted in a bracket with a
minus sign before it.
Step2:Open the parenthesis by changing the sign of each term inside the parenthesis from + to - and
from - to +
Step3: Add the like terms.

Example: Subtract $a^{3}$ + 4a - 2$a^{2}$ - 2 from 3$a^{2}$ + a - 2$a^{3}$ + 3
horizontally
Solution: We have,

3$a^{2}$ + a - 2$a^{3}$ + 3 - ($a^{3}$ + 4a - 2$a^{2}$ - 2)
= 3$a^{2}$ + a - 2$a^{3}$ + 3 - $a^{3}$ - 4a + 2$a^{2}$ + 2
= (- 2$a^{3}$- $a^{3}$) + (3$a^{2}$ + 2$a^{2}$) + (a - 4a) + (3 + 2)
= (- 2 - 1) $a^{3}$ + (3 + 2) $a^{2}$ + (1 - 4) a + 5
= - 3$a^{3}$ + 5$a^{2}$ - 3a + 5 Answer.

The distributive property is used to multiply a Polynomial by a monomial.

Multiplying Polynomials by Monomials

The distributive Property is extended to multiplying two polynomials. Each term of a polynomial is distributed over (or multiplied with) every term of the other polynomial.

Multiplying Polynomials by Monomials Example

This can also be illustrated in a Table. The terms of the polynomials are represented as row and column Headers. Each cell contains the product of terms corresponding to the row and column of the cell.

Product of Polynomials

There are mainly two methods of dividing polynomials. The first method is called the Long division and the second method is called the synthetic method.

Long division method: is very similar to the usual division. The only difference is that the variables are involved in polynomial division.

Consider an example:

Divide P(x) = x4 + 2x3 + 2x + 4 by (x+2)
The demonstration is below.

Dividing Polynomial Example

Quotient = x3 + 2 and Remainder = 0

The synthetic division is a method which uses coefficients only. Here we omit the variables and just consider the coefficients.

First, we write only the coefficients. Since we are dividing by x + 2 we write –2 outside. Write the first number as it is below.
Dividing Polynomial Example using Synthetic Division step1
Now we multiply 1 by -2 and write below the 2 and add. The sum is written next to 1
Dividing Polynomial Example using Synthetic Division step2
Now we multiply 0 by -2 and write below the 0 and add. The sum is written next to 0
Dividing Polynomial Example using Synthetic Division step3
The process is repeated till the last number.
Dividing Polynomial Example using Synthetic Division step4
The last number will the remainder
Here the remainder is 0
To obtain the quotient we use the sums to the left of remainder
The quotient = 1x3 + 0x2 + 0x + 2 = x3+2

The value of x, which satisfies the polynomial equation is called the zero of the polynomial.
That is if x=a is a zero of the polynomial, the P(a)=0.
There are different methods of finding the zero of a polynomial.
Lets consider an example:

Consider a simple trinomial P(x) = x2 - 10x + 24

First method of finding the zero is graphing the polynomial
So lets graph the polynomial. The graph of the above polynomial is given by

Finding Zeros of Polynomials Example

We can see that the graph meets the x axis at two points x = 4 and x=6. So we can say that the two zeros of the above polynomial are x=4 and x=6.

Second method of finding zeros is factoring the polynomial

The given polynomial is a trinomial. To factor a polynomial, the general method is to find a zero using trial and error method.

In this case we put a value for x(say a) arbitrarily and check whether P(a)=0. If P(a)=0, then x=a is a zero of the polynomial and x-a is a factor of the polynomial.

Lets see how it works.

Our polynomial is P(x) = x2 - 10x + 24
First lets try x = 0 then P(0) = 02 - 10(0) + 24 = 24 $\neq$ 0. So x = 0 is not a zero of the polynomial.
Now lets try x = 1 then P(1) = 12 - 10(1) + 24 = 15 $\neq$ 0. So x = 1 is not a zero of the polynomial.
The order of trial is 0, 1, -1, 2, -2....
We can try the numbers randomly also.
Lets try x = 4 then P(4) = 42 - 10(4) + 24 = 30 $\neq$ 0. So x = 4 is not a zero of the polynomial.
To find the order zero we can either use long division or synthetic method.

The highest power of x in a polynomial is called degree of the polynomial. So in
P(x) = an xn + an-1 xn-1 +...........+a1 x1 + a0 where an $\neq$ 0, the degree of the polynomial is n.

The polynomial of degree 1 is called linear polynomial. A polynomial of degree 2 is called a Quadratic polynomial. With degree 3 the polynomial is called cubic polynomial and a polynomial with degree 4 is called Quartic polynomial.

Example:
In the monomial: 3x2, the highest power of x is 2, so the degree of the polynomial is 2.
In the binomial: 4x + 3, the highest power of x is 1, so the degree of the polynomial is 1.
In the trinomial: 9x2 + 6x + 8, the highest power of x is 2, so the degree of the polynomial is 2.

The simplified form of a polynomial is obtained by combining the like terms in the polynomial. Like terms are terms with the same degree (power).

Lets consider an example:

Consider the polynomial P(x) = x2 + 15x + 2x2 - 10x + 24
Lets simplify this polynomial
Here x2 and 2x2 are like terms. Similarly 15x and -10x are like terms. So we combine them to find the simplified form of this polynomial. The simplified form is given by
P(x) = x2 + 15x + 2x2 - 10x + 24 = 3x2 + 5x + 24

Another example is
P(x) = 2x3 + 25x2 + 2x2 + 30x - 9
= 2x3 + 27x2 + 30x - 9
So to simplify each polynomial, we just have to combine the like terms.

Usually a polynomial is written in the descending order. When the terms in a polynomial are arranged in the increasing order of the degree, then it is said to be in ascending order of the polynomial.

Example:
Consider the polynomial P(x) = 2 + 5x + 9x2 + 36x3 + 3x4
In this polynomial, the terms are arranged in the increasing order of the degree. Hence it is in the ascending order.

Another example is P(x) = -5 -9x + 30x2 + 27x3 + 22x4
In this case also, the terms are arranged in the increasing order of the degree of individual terms. So we can say that this polynomial is in an ascending order.

The normal form of representing a polynomial is using the descending order of terms. When the terms in a polynomial are arranged in the decreasing order of the degree, then it is said to be in the descending order of the polynomial. Descending order form is more used than the ascending order form of a polynomial.

Example:
Consider the polynomial P(x) = 22x4 + 5x3 + 9x - 36
In this polynomial, the terms are arranged in the decreasing order of the degree of each term. Hence it is in a decreasing order.

Another example is P(x) = 22x4 + 27x3 + 30x2 - 9x - 5
In this case also; the terms are arranged in the decreasing order of the degree of individual terms. So we can say that this polynomial is in a descending order.

There are many methods of factoring polynomials. We can factor a polynomial either by grouping or by taking common factors outside or by using identities. The Quadratic polynomials can be factored using the Quadratic formula also.

Lets consider an example for each case:

Factoring By Grouping
Consider the polynomial P(x) = x4 + 2x3 + 2x + 4
In this case we group the first two and the last two terms. Then taking common factors we can simplify.
P(x) = x4 + 2x3 + 2x + 4
= x3 (x + 2) + 2(x + 2)
= (x + 2)(x3 + 2)

Factoring by taking common factors outside
Consider the polynomial P(x) = 4x2 + 4x + 4
In this case we take out the common factor 4.
We cannot simplify it further.
So the polynomial becomes
P(x) = 4x2 + 4x + 4
= 4(x2 + x + 1)

Factoring using identities
There are many algebraic identities. We can factor polynomials using identities like
a2 - b2 = (a + b)(a - b)
a2 - 2ab + b2 = (a - b)2
a2 + 2ab + b2 = (a + b)2
There are many identities, we have just quoted three of them.

Lets consider the following example.
Consider P(x) = x2 - 25
Here the polynomial is of the form a2 - b2 = (a + b)(a - b).
So we can factor the polynomial as P(x) = x2 - 25 = (x + 5)(x - 5)

Factoring using quadratic formula
This method can be used only for quadratic polynomials and polynomials which can be reduced to quadratic polynomials.

If P(x) = ax2 + bx + c is a polynomial, quadratic formula is
x = $\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$

Consider the polynomial P(x) = x2 - 10x + 24
Comparing with the standard polynomial P(x) = ax2 + bx + c
Here a = 1, b = -10 and c = 24
Then using quadratic formula,
x = $\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$
= $\frac{-(-10) \pm \sqrt{(-10)^{2} - 4 \times 1 \times 24}}{2 \times 1}$
= $\frac{10 \pm 2}{2}$
= 4, 6
Since x = 4 is a zero of the polynomial, (x - 4) is a factor. Similarly, since x = 6 is a zero of the polynomial, (x - 6) is a factor
So the factors are (x - 4)(x - 6)

Common factors are factors that are common to all the terms in a polynomial. The biggest factor that is common to all terms is called the greatest common factor (GCF).

Lets consider an example

Consider the polynomial P(x) = 10a - 15b + 25 here 5 is common to all terms. So we can say that the common factor is 5.

The polynomial when added to the given polynomial gives a sum 0 and this is called the additive inverse of a polynomial.

Consider the polynomial P(x) = x4 - 2x3 - 3x + 4 . The additive inverse of this polynomial is given by - P(x) = - x4 + 2x3 + 3x - 4 . We just have to change the sign of each term in the polynomial.

Lets consider another polynomial P(x) = x3 + 5x + 9 . The additive inverse of this polynomial is given by -P(x) = - x3 - 5x -9 . Here also we have just changed the sign of each term in the polynomial.