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Polynomials are widespread in maths. They help to reduce complex problems into the form of letters and numbers, that are easy to manipulate and work with. You will find them frequently in your maths textbooks. In fact, polynomials form an essential part of algebra. Without understanding the concept of polynomials, it is hard to move ahead in complete understanding and application of mathematics.

## What is a Polynomial?

The word polynomial is an amalgamation of a Greek word, “poly” (meaning many) and Latin “nomen” (meaning name). Thus, the word “polynomial” means “many names”. As you can see that it is hard to make anything out of the literal meaning of the word.

## A little bit of History

The roots of the word “polynomial” date back to 17th century when it was first used. The problems that we now represent as polynomial equations were known and even solved in the ancient era as early as 250 AD. Later, during the “Dark Age” of the Europe, eminent Muslim scientists such as Al-Khowarizmi (800-847 AD), Al-Karaji (953-1029) and Al-Samawal (1130-1180) invented and applied the concept of notations to perform different calculations.

In 16th century A.D., the solution to the polynomial equations with degree higher than two was accomplished by European mathematicians including Cardano (1501-1576) and Ferrari (1522-1565). World renowned mathematician Newton (1643-1727) also contributed a lot in this field. A number of advancements were carried out by several mathematicians and physicists after then.

## Definition

A polynomial is a mathematical statement that consists of variables and real numbers (constant and coefficients). The variables are raised to whole number exponents/powers.

For example, the given expression is a polynomial of one variable (x) only.

$2x^{2}+3x+1$

Variables (or Indeterminates): The entities that can take on value of any real number are known as variables. They act as placeholder in mathematical equations. The letters x, y and z are generally used to represent the variables. They are also called Indeterminates since they do not have a specific value.

Coefficients: In simple words, the real numbers that appear with the variables are called coefficients. In the polynomial $3x^{3}+2x+8$, 3 and 2 are the coefficients.

Constants: The real numbers that appear standalone (without the variable) are called constants. In the polynomial $3x^{3}+2x+8$, only the number 8 is a constant.

There are certain aspects that characterize the polynomials.

One, that the variables are raised to only positive exponents. Therefore, $x^{-2}+3$ is not a polynomial since it contains negative exponent of the variable x.

Two, that the exponents must be whole number. Therefore, $x^{\frac{1}{2}}-2x$ is not a polynomial since one of the exponents i.e., $\frac{1}{2}$ is not a whole number. Note that constants as well as coefficients are not mandated to be whole numbers.

Three, that there can be more than one variable in the expression of the polynomial. For example, $x^{2}+2y+z^{3}$ is a polynomial of three variables.

Exercise: Identify which among the following expressions are polynomials. Mark and classify the variables, coefficients and constants in the polynomials.

• $\frac{1}{x}+2y$
• $y^{2}+7\sqrt{x}$
• $x-y-z$
• $x^{50}+x^{3}+2$
• $x$
• $\frac{1}{2}+6x$

Interesting fact: A constant e.g., 3 can also be referred as a polynomial. It is known as a constant polynomial.

From now on, in this article, by ‘polynomial’ we would mean ‘a polynomial of single variable’.

## Degree of a polynomial

The highest power to which a variable is raised in a polynomial is called the degree of the polynomial. For example, the expression ‘2x+1’ is a polynomial of degree 1. The expression ‘$x^{3}+3$’ is a polynomial of degree 3. In mathematics, it is a good practice to write the term with the highest degree first (on the left), then the lower degree term and so on. Constants are always written at the last.

Exercise: Find the respective degrees of the following polynomials.

• $x^{100}+x^{99}+1$
• $x^{2}+\frac{1}{2}$
• $0.6x$
• $23$

Note that a polynomial consisting of a single constant can be considered a polynomial of degree 0.

Notation: The polynomial of degree ‘n’ is denoted as $P_{n}(x)$. For example, a general polynomial of degree 2 is represented as

$P_{2}(x) = ax^{2}+bx+c$

Polynomial function: It is a function (a function is a mathematical expression that outputs only a single value for a given input) that involves variable with only positive integer exponents. The examples of a polynomial function include:

• $f(x)=2x+3$
• $f(x) = x^{2} + 3x+1$
• $f(x) = x^{909} + 10$

A function is not always a polynomial. Examples of expressions that are functions but not polynomials include:

• $f(x) = 4x^{\frac{1}{2}}$ (or $P(x) = 4\sqrt{x}$)
• $f(x) = 1+2x^{-1}$
• $f(x) = \frac{2}{x^{2}} + \frac{1}{x} + 1$

Exercise: Find among the following the expressions that, you suspect, are polynomial functions.

• $\frac{1}{10}x+1$
• $\frac{10}{\sqrt{x}}+1$
• $x\sqrt{10}+1$
• $\frac{x^{2}}{\sqrt{10}}+100$

The polynomials can be classified on the basis of their degree. They can be classified as:

Zero-degree polynomials (Constant polynomials): As the name suggests, they have 0 as the highest degree of the variable (or indeterminate). They are generally represented as:

$P_{0}(x) = ax^{0}$

You may now see why they are also called the constant polynomials since they are equal to a constant ($x^{0}=1$). An example of a constant polynomial is $P_{0}(x) = 3$.

First-degree polynomials (Linear polynomials): These polynomials have variables (or indeterminates) with the highest exponent of one. They are generally represented as:

$P_{1}(x) = ax + b$

The reason why they are called linear polynomials is that they have a linear relation between the input and the output. It will be elaborated in detail when we discuss the graphs of the polynomials later. An example of a linear polynomial is $P_{1}(x) = 2x + 1$. Note that the constant b can be 0 but a should be non-zero (Can you explain why a should be non-zero?).

Second-degree polynomials (Quadratic polynomials): These polynomials have variables (or indeterminates) with the highest exponent of two. They are generally represented as:

$P_{2}(x) = ax^{2}+bx+c$

An example of quadratic polynomial is $P_{2}(x) = 2x^{2}+3$. Note that b and c can be 0 but a should be non-zero.

Third-degree polynomials (Cubic polynomials): These polynomials have variables (or indeterminates) with the highest exponent of three. They are generally represented as:

$P_{3}(x) = ax^{3}+bx^{2}+cx+d$

An example of a cubic polynomial is $P_{3}(x) = x^{3}+5x^{2}+10x+40$. Note that the numbers b, c and d can be 0 but not a.

Similarly, the higher degree/order polynomials are named accordingly. It should be noted that the above discussed polynomials also qualify as polynomial functions (You can verify it yourself). Next, we will see how these polynomials take interesting shapes as we graph them.

## Graphs of polynomials

Now it is the right moment that we graph the polynomial functions we have discussed earlier. First one is the zero-degree polynomial. It is graphed as

Figure 1 Graph of $P_{0}(x) = 9$

The vertical axis is the one along which we plot the computed value of the function at a given value of x. It is not difficult to understand why the graph looks like this. For any value of x we have only one value i.e., 9 as output.

Next is the graph of the linear polynomial $P_{1}(x) = 2x+1$.

We see that the y-intercept (the point on vertical axis where the graph intersects) of this polynomial is 1. It can also be seen that x-intercept (the point on the horizontal axis where the graph intersects) is -0.5. You can also evaluate the x and y intercepts by replacing $P_{1}(x)$ with 0 (to find x-intercept) and by replacing x = 0 (to find y-intercept).

Exercise: Graph the polynomial function $P(x) = 10x-4$. Find its x and y intercepts also.

The graph of a quadratic polynomial, for example, $P_{2}(x) = x^{2}+2x-1$ is interesting.

You can see it intersecting the x-axis at two different points. It can also be seen that if forms a V-shape. Quadratic polynomials will be discussed in great detail later.

Finally, the graph of a cubic polynomial $3x^{3}-2x^{2}+x-1$ is given as:

Compare the graphs of 2nd and 3rd degree polynomial and observe the difference.

As discussed earlier, the quadratic polynomial is any polynomial having degree of 2. Quadratic polynomial is also called Quadratic function.

### Arithmetic operations between quadratic polynomials

Arithmetic operations can be performed between two or more quadratic polynomials (any degree polynomials in general). They can be added, subtracted, multiplied and divided by other polynomials. For example, given we have two quadratic polynomials $q(x)=x^{2}+3x+2$ and $r(x)=x^{2}+4x$, we can have:

• $q(x) + r(x) = x^{2}+3x+2+(x^{2}+4x)$
• $q(x) + r(x) = 2x^{2}+7x+2$

2) Subtraction

• $q(x) - r(x) = x^{2}+3x+2-(x^{2}+4x)$
• $q(x) - r(x) = -x+2$

3) Multiplication

• $q(x) r(x) = (x^{2}+3x+2)(x^{2}+4x)$
• $q(x) r(x) = x^{4}+7x^{3}+14x^{2}+8x$

4) Division

• $\frac{q(x)}{r(x)}=\frac{x^{2}+3x+2}{x^{2}+4x}$

In case of addition, subtraction and multiplication, we always get another polynomial whereas in case of division, we may or may not get another polynomial. For example, in the above example, you can see that the first three operations result in another polynomial, while the operation of division did not yield a polynomial. If we have another polynomial s(x) = 2x, then we can see that $\frac{r(x)}{s(x)}=\frac{x^{2}+4x}{2x}=\frac{x}{2} + 2$, that is also a polynomial.

Exercise: Perform the four arithmetic operations on the following quadratic polynomials. Also find the degree of the resulting polynomial (if applicable).

• $q(x)=2x^{2}+5x, \, r(x)=x^{2}-1$
• $q(x)=\frac{1}{1}x^{2}, \, r(x)=x^{2}+2x+4$

A zero of a quadratic polynomial p(x) is the value of the variable x for which the polynomial p(x) is equal to 0. Zeroes of a quadratic polynomial p(x) are the roots of the equation p(x)=0. Given the general expression $p(x)=ax^{2}+bx+c$, the zeroes of p(x) are the roots of the equation $x^{2}+bx+c=0$.

It should be noted that ‘root’ is technically same as ‘solution’.

For example, the zeroes of the quadratic polynomial $p(x)=2x^{2}+5x+2$ are basically the solutions of the equation $2x^{2}+5x+2=0$. As a matter of fact, a quadratic polynomial has two zeroes. The zeroes of the aforementioned quadratic polynomial come out to be $-\frac{1}{2}$ and -2. In other words, the value of polynomial goes zero at $x = -\frac{1}{2}$ and x=-2 (Try to verify it by replacing x with these values in the expression of the polynomial).

It is worth nothing that zeroes of a quadratic polynomial are not always real. They may come out to be complex numbers.

Exercise: Find the zeroes of the quadratic polynomial $x^{2}-1$. You can try to guess the answer by plugging in different values of x in the expression. Or you can solve for x in $x^{2}-1=0$.

There is a well-established system of finding roots of a quadratic polynomial in mathematics. The famous quadratic formula is used for this purpose. Quadratic formula is given by:

$x_{1,2}=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

where, $x_{1,2}$ are the two roots.

The reader should not be worried about the parameters a, b and c. They represent constants and coefficients of the general quadratic polynomial $ax^{2}+bx+c$.

Let us implement the quadratic formula in solving an example. Given the quadratic polynomial $p(x)=2x^{2}-2x-3$. The zeroes of the polynomial (or the roots of the equation px=0), can be found by first comparing p(x) with the standard quadratic polynomial. After comparing, we get:

• a = 2
• b = -2
• c = -3

The next step is simply putting the parameters in the quadratic formula and evaluating the results. Thus, we have

$x_{1,2}=\frac{-(-2)\pm \sqrt{(-2)^{2}-4(2)(-3)}}{2(2)}$

which yields the two zeroes at 1.82288 and -0.822876. In other words, at x = 1.82288 and -0.822876, the given polynomial equates to 0.

Quadratic formula is a powerful and widely used tool in mathematics even at advanced levels. The students are therefore advised to learn it by heart.

Exercise: Using the quadratic formula, find the zeroes of the polynomial $p(x)=x^{2}-1$.

Discriminant: The term ‘$b^{2}-4ac$‘ in the quadratic formula is given the name, discriminant. You can estimate its importance by considering the fact that it determines the nature of the zeroes of the quadratic polynomial.

If discriminant is positive, then zeroes are real and distinct. If it is equal to zero, the zeroes are real and repeated (equal). Or if it is negative, the zeroes are complex and distinct.

Let us take a look at the graph of the polynomial $p(x)=2x^{2}-2x-3$ to see how the concept of ‘zeroes of quadratic polynomial’ arise visually.

Figure 5 Graph of $p(x)=2x^{2}-2x-3$

You can see that the zeroes of the polynomial are simply the values of x where the curve intersects the x-axis. It is easy to see that if we shift the curve up gradually, there comes a moment when only the tip of the V-shaped curve intersects with the x-axis. That occurs when discriminant is zero. Also, since the curve will be intersecting at one point only, the zeroes will be real and repeated/equal.

It is also possible that the curve never intersects the real x-axis at all. That happens when discriminant is negative and the zeroes come out to be complex.

Some quadratic polynomials can be represented as products of other polynomials, which are called their factors. For example, the polynomial $x^{2}+3x+2$ can be represented as (x+1)×(x+2). Thus, these first order polynomials (x+1) and (x+2) are factors of the quadratic polynomial $x^{2}+3x+2$.

Different methods for factoring a quadratic polynomial have been devised by mathematicians. We will discuss two of them.

Splitting the x-term: Given the general form of a quadratic polynomial $p(x)=ax^{2}+bx+c$, this method deals with the manipulation of the parameter b with respect to the parameters a and c. In this method the goal is to split the term bx as sum of two individual terms (say $b_{1}x+b_{2}x$) such that the product of the coefficients of the two is equal to the product ac i.e., $b_{1}b_{2}=ac$ The rest of the steps follow naturally.

For example, we have the polynomial $p(x)=4x^{2}-9x+2$. We have to split the term ‘-9x’ into sum of two terms such that the product of the coefficients of the two is equal to ac = 4 (2) = 8. The sign of the term ‘-9x’ should also be kept in mind.

It can be observed that the polynomial can be rewritten as $p(x)=4x^{2}-8x-x+2$. This expression satisfies the requirements i.e., $b_{1}b_{2}=(-1)(-8)=8=ac$. Now the next step is to take out common factors as

$4x^{2}-8x-x+2 = 4x(x-2)-1(x-2)$

It is worth realizing that the goal is to factor out the terms such that we have the same linear polynomial enclosed in brackets with both the terms i.e., (x-2) in this case. This helps factoring it out. Now, we have a common factor (x-2). Thus, the expression reduces to

$4x(x-2)-1(x-2) = (x-2)(4x-1)$

Exercise: Try to factor out the following quadratic polynomials using the method discussed above.

• $5x^{2}-9x-2$
• $x^{2}+2x+1$

Application of Quadratic formula: This one is interesting. You may recall that by using the quadratic formula we can find the two roots of the equation p(x) = 0, where $p(x)= ax^{2}+bx+c$. The two roots that we get are symbolically represented as $x_{1}$ and $x_{2}$ (Note that both are numbers not variables).

The next step is to build the linear polynomials (i.e., the factors of the quadratic polynomial). This can be done simply by writing the roots as:

$x = x_{1}$ and $x = x_{2}$

These can be rewritten as

$x- x_{1}=0$ and $x- x_{2}=0$

And ‘ta da’, you have the two factors. The original quadratic polynomial can be then written as:

$ax^{2}+bx+c=(x-x_{1})(x-x_{2})$

Let us take a look at an example. A polynomial $p(x)=x^{2}-2x-2$ is given. Using the quadratic formula, we find the zeroes of the p(x) (or roots of p(x) = 0). They are $x_{1}=2.73205$ and $x_{2}=-0.73205$. Thus, the polynomial can be expressed in factored form as:

$x^{2}-2x-2 = (x-2.73205)1(x+0.73205)$

Caution: Note that the factors are written as x minus $x_{1}$ and x minus $x_{2}$”. Therefore, you should be cautious of the negative sign while expressing the polynomial in form of factors.

It was stated earlier that some quadratic polynomials can be expressed in form of factors. But as you can see that by the application of quadratic formula, any quadratic polynomial can be factored. Then why the discrepancy? Well, a quadratic polynomial can have complex zeroes.

And, if we are bound to consider only the real zeroes then we cannot have factors involving complex numbers.

However, in broad sense, polynomials can have factors involving complex numbers as well.

Exercise: Find the factors of the following polynomial using the quadratic formula.

$-\frac{1}{2}x^{2}-2x-2$

## More on the graphs of Quadratic polynomial

We have already observed what the graph of a quadratic polynomial looks like. In this section we explore how to modify/transform the graphs, and how the coefficients and the constant affect the graph.

Transformation: Four basic transformations can be performed on basically any type of polynomial. They are:

• Translation/Shifting along vertical axis (y-axis)
• Translation/Shifting along horizontal axis (x-axis)
• Scaling along vertical axis
• Scaling along horizontal axis

Let us take a polynomial $p(x)=-x^{2}-2x+2$. One by one, all of the transformations will be performed and depicted graphically.

It is apparent that the shifting of the graph along y-axis can be done by adding/subtracting a number with the polynomial. Adding a positive number shifts the graph up and vice versa.

The graph has been shifted down by 4 units (Every point translates 4 units down!). Next, we translate the curve 4 units along the x-axis. You will observe that shifting along the horizontal axis is counter-intuitive (in respect of direction of translation).

Indeed, it shifted towards right for p(x minus 4). How would you shift it 1.5 units left?

Similar is the case for scaling along the two axes. Multiplying the polynomial with a number greater than 1 expands it along the y-axis and vice versa. On the other hand, multiplying the variable x by a number greater than 1 compresses the curve along the x-axis and vice versa.

Note that the curve compressed by a factor of 4. Also note that the center of compression is y-axis (not the central axis of the p(x)!).

Significance of coefficients and constant: There are three parameters in the expression of a quadratic polynomial (two coefficients and a constant). Using the general expression of a quadratic polynomial $p(x)= ax^{2}+bx+c$, we have two coefficients a and b and a constant c.

Let us see the effect of each on the graph of a quadratic polynomial. Starting with the parameter a, we vary it from negative values to positive.

Wow! There is so much to observe and learn from this graph. Start with the negative values of a, at a = -10 (the green curve). It is noticed that for negative values of a, the curve appears as an inverted V-shape. The curve for a = -10 is more ‘pointed’ and less wide as compared to the curve for a = -1. Then at a = 0 (although the quadratic polynomial reduces to a linear polynomial at this point), we observe a straight line. The curve then flips its direction and starts folding for a > 0. Greater the value of a, more pointed the curve becomes.

Fun exercise: Imagine an animated film where you have the parameter a varying from negative infinity to positive infinity for the above-mentioned polynomial. How will the curve look as the parameter a varies?

There are many other aspects of the graph you should discover yourself.

Next, we vary the parameter b.

Figure 11 Graph of $x^{2}+bx+1$ for different values of b

To observe what is going on, use the same technique as before. Start with the curve for which b = -3, then look how the graph for the polynomial varies as b increases. Note that at b = 0, the polynomial reduces to $x^{2}+1$ which is still a quadratic polynomial!

As far as the variation of c is concerned, you should be able to guess what should happen. Yes, that is right. The curve will translate along y-axis.

Figure 12 Graph of $x^{2}+x+c$ for different values of c

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