Introduction
 Polynomials are mathematical expressions with one or more terms separated by an addition or subtraction operator.
 In a polynomial, the variables are raised to a nonnegative integer power.
 For example, ${x}^{2}+7x+10$ is a polynomial in the variable $x$ with three terms.
 The highest power of the variable in any polynomial is called its degree.
 In the above example, the degree of the polynomial is two.
 Note: A polynomial with a single term is called a Monomial.
Addition and Subtraction of Polynomials
 The addition & subtraction of polynomials can be carried out in the following two steps:

 Identify like terms in the polynomial
 Combine them according to the correct integer operation (addition or subtraction)
 Note that like terms have the same variables raised to the same exponents.
 The addition or subtraction of polynomials results in a polynomial of the same degree.
Polynomial Multiplication
 To multiply a polynomial by a monomial, apply the distributive property and simplify each term.
 For example: $2x\xb7\left(x5\right)$$=2x\xb7x2x\xb75$$=2{x}^{2}10x$
 Likewise, to multiply two polynomials, we distribute each term of the one polynomial to all the terms of the other polynomial (FOIL method).
 For example: $\left(5x2\right)\xb7\left(x+3\right)$$=5x\left(x+3\right)2\left(x+3\right)$$=5{x}^{2}+15x2x6$$=5{x}^{2}+13x6$
 Note: Two or more polynomials, when multiplied, always result in a polynomial of a higher degree (unless one of them is a constant polynomial).
Dividing Polynomials
 To divide a monomial by a monomial, divide the numerical coefficients, divide the variables separately, and then multiply the results.
 Example: $\frac{14{x}^{6}}{7{x}^{2}}$$=\frac{14}{7}\xb7\frac{{x}^{6}}{{x}^{2}}$$=2{x}^{4}$
 To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
 Example: $\frac{8{m}^{5}+6{m}^{3}3{m}^{2}}{2m}$$=\frac{8{m}^{5}}{2m}+\frac{6{m}^{3}}{2m}\frac{3{m}^{2}}{2m}$$=4{m}^{4}3{m}^{2}+\frac{3m}{2}$
 The long division method is used to divide a polynomial by a polynomial.
Solved Examples
Question 1: Add the following polynomials: $\left(5{m}^{3}{n}^{2}\right)+\left(2{m}^{3}6{n}^{2}\right)$.
Solution: $\left(5{m}^{3}{n}^{2}\right)+\left(2{m}^{3}6{n}^{2}\right)$$=5{m}^{3}{n}^{2}+2{m}^{3}6{n}^{2}$$=5{m}^{3}+2{m}^{3}6{n}^{2}{n}^{2}$$=7{m}^{3}7{n}^{2}$
Question 2: Find the product: $\left(m5\right)\xb7\left(3m+6\right)$.
Solution: $\left(m5\right)\xb7\left(3m+6\right)$$=m\xb7\left(3m+6\right)5\xb7\left(3m+6\right)$$=3{m}^{2}+6m15m30$$=3{m}^{2}9m30$
Question 3: Find the product: ${\left(5x\right)}^{2}$.
Solution: ${\left(5x\right)}^{2}$$=\left(5x\right)\xb7\left(5x\right)$$=5\xb7\left(5x\right)x\left(5x\right)$$=255x5x+{x}^{2}$$=2510x+{x}^{2}$
Question 4: Divide $\left(18{x}^{7}+4{x}^{5}2{x}^{2}\right)$ by ${x}^{2}$.
Solution: $\frac{18{x}^{7}+4{x}^{5}2{x}^{2}}{{x}^{2}}$$=\frac{18{x}^{7}}{{x}^{2}}+\frac{4{x}^{5}}{{x}^{2}}\frac{2{x}^{2}}{{x}^{2}}$$=18{x}^{5}+4{x}^{3}+2$
Question 5: Find the degree of the polynomial ${x}^{3}2{x}^{2}+4$.
Solution: The highest power of the variable in the given polynomial is three; hence, its degree is three.
Cheat Sheet
 To add or subtract the polynomials, add or subtract the like terms.
 To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial.
 The "FOIL" method is used to multiply two binomials.
 To divide a polynomial with a monomial, divide each polynomial term by the monomial.
 To divide two polynomials, use the long division method.
 Below are some special product rules:

 ${\left(x+y\right)}^{2}={x}^{2}+2xy+{y}^{2}$
 ${\left(xy\right)}^{2}={x}^{2}2xy+{y}^{2}$
 $\left(x+y\right)\left(xy\right)={x}^{2}{y}^{2}$
 ${\left(x+y\right)}^{3}={x}^{3}+3{x}^{2}y+3x{y}^{2}+{y}^{3}$
 ${\left(xy\right)}^{3}={x}^{3}3{x}^{2}y+3x{y}^{2}{y}^{3}$
 $(x+y)({x}^{2}xy+{y}^{2})={x}^{3}+{y}^{3}$
 $(xy)({x}^{2}+xy+{y}^{2})={x}^{3}{y}^{3}$
Blunder Areas
 Only like (similar) terms of polynomials can be added or subtracted.
 One must be careful while multiplying polynomials involving negative signs.
 Abhishek Tiwari
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