What is a Polynomial? 

Polynomial is an algebraic expression with a non-negative integer exponent for variables. Suppose that there are 10 students in a class. If each one is given 2 pencils, then what is the total number of pencils the entire class has? To find the total value we can either add 2 ten times or multiply 2 and 10. Suppose 2 new students join the class. Now the total number of pencils is 24. If 5 more students join, the total number of pencils is 34. The number of students in each case is varying. So, we can mention it generally as $2x$ where $x$ is the number of students. Here $x$ is called variable and 2 is a constant and since it is multiplied with $x$ it is called the coefficient of $x$. A term that does not change is called a constant. In $2x+4$ , 4 is constant. $2x+4$ is an algebraic expression, and it can be added or multiplied. Some terms like $-x$, 10$x$, $\frac{1}{2}x$ are also algebraic expressions. Polynomial is one of these kinds.

A Polynomial with One Variable

When the number of students is increasing or decreasing, we represent it by the letter $x$, calling it a variable. Even constants can be represented by any letter. Instead of writing $2x$, it can be written as ‘ax’ where $a$ is the constant. Here ‘2’ is called the coefficient of the variable $x$. Here, ‘$a$’ will not change, and hence both the letters have two different values and meanings. The perimeter of a square is 4 when multiplied by the length. If the length is 7, the perimeter is 28. The length can be anything. In general, the perimeter is $4x$ where $x$ can take any positive value in this case. But generally, $x$ can take any real value.

Exponents

Now let us see the area of a square. The value is found out by multiplying length with length. That is, $x\times x=x^2$. In algebra, the number two means that the variable x is multiplied twice. If x is multiplied thrice, it is denoted as $x^3$. Here, 2 and 3 are called exponents of the variable x. The exponents should always be a whole number for polynomials. Here,$x^2$ is of degree 2 since the power of the variable x is 2 and $x^3$ is of degree 3.

Constant Polynomial 

The term 2 is called a constant polynomial. Here the variable x has degree 0 and the value of $x^0$ is 1. It can also be written as 2$x^0$. Similarly, -3, 8, 19 are also called constant polynomials. 0 is called a zero polynomial.Is $x^{-1},\sqrt[3]{x},x^{\frac{3}{5}}$ a polynomial? No, because the exponents are not whole numbers.

Degree of a Polynomial

Consider the expression $x^2+3x+1$ . This is a polynomial because the exponents are whole numbers. There are 3 terms added up. They are $x^2$, $3x$ and 1. The degree of $x^2$ is 2, the degree of 3$x$ is 1, and the degree of 1 is 0 (1 can be written as 1$x^0$ ). The highest degree is the exponent of $x$ with the greatest whole number of all. Here it is 2. Here, 2 is the degree of the polynomial. Now find out for expression, $x^4+9x^3+2$ . The answer is 4. Look at the previous steps and find out how it is 4. If the degree of the polynomial is two, it is called quadratic. For example, $a{x}^2+bx+c$ , where $a$, $b$ and $c$ are constants, and they can be any real number. The constant a cannot be zero because if a is zero, then the first term vanishes (zero multiplied with anything is again zero). So the highest degree is not 2 anymore. Then it is not quadratic. So, it is expressed as $a\ne 0$ for a quadratic polynomial.

If the highest degree is 3, it is called cubic. For example, $a{x}^3+b{x}^2+cx+d$, where the constant a cannot be 0. Why? It is just like the above case.

Zeroes or Roots of a Polynomial 

Let us try substituting different values for the variable $x$ in $p\left(x\right)=x+3$.

When $x=1$, $p\left(1\right)=1+3=4$.

When $x=0$, $p\left(x\right)=0+3=3$.

When $x=-1$, $p\left(x\right)=-1+3=2$.

What is $p\left(x\right)$ at $-3$?

Here,

$p-3=0$

$-3$ is the zero or root of $p\left(x\right)=x+3$.

Now can you find the root of $p\left(x\right)=x-9$?

(The answer is $9$).

Let's now try substituting $0$ in the place of the variable $x$ in the expression $x^2+3x+2$.

This is a quadratic polynomial $p\left(x\right)$ and can be written as

$p\left(x\right)=x^2+3x+2$

Replacing $x$ with $0$,

$p\left(0\right)=0^2+3\times 0+2$

$p\left(0\right)=0+0+2$

$p\left(0\right)=2$

The result is 2. Can you find a number which on substituting in the place of $x$ equates $p\left(x\right)$ to $0$?

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check. 

$p(-2)={(-2)}^2+3\times (-2)+2$

$p(-2)=4+(-6)+2$

$p(-2)=-2+2$

$p(-2)=0$

So, at $x=-2$, the equation is 0. This is called zero or root of the equation $p\left(x\right)$.

Can you find any other number other than $-2$? What about $-1$? It also equates $p\left(x\right)$ to 0. This is also a root for the equation $p\left(x\right)$. Can there be any more roots for $x^2+3x+2$ other than -2 and -1? No. Because the degree is 2, only 2 roots can be derived.

Try to find the zeroes of $p\left(x\right)$ where $p\left(x\right)=$$x^2+2x-15$.

Hint: Substitute -5 and 3 in the place of the variable $x$. 

But this is a very long process as you may have to substitute many values and wait for 0 to arrive. Try equating the polynomial to zero and find the root.

If $p\left(x\right)=x-11$ , equating it to 0 gives $x-11=0$ Now add 11 on both sides. It is $x=11$.

So, 11 is the root of$p\left(x\right)=x-11$.

For quadratic polynomials, use the factorization method. In $p(x)=x^2+3x+2$, the coefficient of the middle term $x$ is 3.

Write 3 as the sum of 2 numbers in a way that when you multiply the two numbers you should get the constant term 2.

The two numbers are 2 and 1 because when you multiply 2 and 1 you will get 2 (the constant term $p\left(x\right)$).

So, $x^2+3x+2$ is written as $x^2+2x+x+2$.

Now,

$\begin{array}{c}{x}^2+2x+x+2=x(x+2)+(x+2)\\ \end{array}$= $(x+1)(x+2)$.

Equating this to 0 we get, $\left(x+1\right)\left(x+2\right)$ $=0$.

$\left(x+1\right)\left(x+2\right)$ becomes 0 when $x=-1$ or $x=-2$. -1 and -2 are the roots.

For the middle terms which cannot be written as the sum of 2 numbers where on multiplication it equals to the constant term, we can use the formula for finding the roots of the quadratic equation:

$\frac{-b+\sqrt{b^2-4ac}}{2a},\frac{-b-\sqrt{b^2-4ac}}{2a}$

Here $a$ is the coefficient of $x^2$, $b$ is the coefficient of $x$ and $c$ is the constant term.

Formula

The general form of a polynomial of degree $n$ in one variable is

$p\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_{1}x+a_0$

Where $n$ is a non-negative integer, $a_0$, $a_1$, $a_{n-1}$, and $a_n$ are real numbers, and $a_n\ne 0$.

The quadratic formula to find zeroes of quadratic polynomials of the form

$p\left(x\right)=a{x}^2+bx+c$ ($a\ne 0$) is

$\frac{-b+\sqrt{b^2-4ac}}{2a},\frac{-b-\sqrt{b^2-4ac}}{2a}$

Where, $a$, $b$, and $c$, are real numbers such that a is non-zero.

Context and Applications 

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for:    

• Bachelor of Science in Mathematics        
• Master of Science in Mathematics