## Polynomials and Zeroes of polynomials

This is the 1st part of the four part series on Polynomials. It complements the section corresponding to Exercise 1 of Chapter 2 of NCERT Class 10 Maths.

In this session we will discuss,

- What are polynomials,
- Degree of polynomials,
- Value(s) and Zeroes of polynomials, and,
- Geometric representation of a polynomial and meaning of its Zeroes.

Discussions are divided into three sections:

**First:** concepts and methods explained in an easy to understand way,

**Second:** example problems used for establishing the concepts and methods clearly and,

**Third:** the solutions to the problems in the 1st exercise of NCERT Maths Class 10 Chapter 2.

### What are polynomials and degree of a polynomial

A polynomial is an algebraic expression in a single variable with positive integer powers of the variable in each term of the expression (that is either added or subtracted together).

#### Examples of polynomials

$7x+3$, $x^2+2x-8$, $x^3-3x+7x+21$ are all polynomials in single variable $x$.

The first is a *linear polynomial of degree 1*, the second is a *quadratic polynomial of degree 2* and the third is a *cubic polynomial of degree 3*.

**Degree of a polynomial** is the highest power of $x$ in the polynomial expression.

#### Examples of expressions that are not polynomials

$\displaystyle\frac{1}{x-1}$, $\sqrt{x}+7$, $\displaystyle\frac{x^2}{x^3-4x+5}$ are not polynomials, though these are algebraic expressions in single variable $x$.

In a polynomial in $x$, each term added to form the expression must be of the form, $\alpha x^{\beta}$ where coefficient $\alpha$ must be a real number (positive, negative, rational fraction or even irrational) and power $\beta$ must be a positive integer ($\beta \geq 0$).

#### Short form of a polynomial

In short form, a polynomial in $x$ is expressed as, $p(x)$.

For example, the polynomial $x^2+2x-8$ is expressed in short form as,

$p(x)=x^2+2x-8$.

#### General form of polynomials

In general, a polynomial in $x$ of degree 1, 2 and 3 are expressed respectively as,

$ax+b$, $ax^2+bx+c$, and $ax^3+bx^2+cx+d$.

$a$, $b$, $c$ and $d$ are the **coefficients** of the terms in $x$ (the power of $x$ for the numeric term $d$ is 0).

A **general polynomial** may be of degree $n$ where $n$ is a non-zero positive integer.

We will be concerned only with polynomials of degree 1, 2 and 3 out of which quadratic polynomials are of most interest.

### Value and Zeroes of a polynomial

The short form $p(x)$ helps us to express the value of a polynomial.

For example, at $x=-3$, the value of the polynomial $x^2+2x-8$ is evaluated by substituting $-3$ as value of the single variable $x$ in the polynomial. The result obtained is,

$p(-3)=(-3)^2-6-8=-5$.

At $x=-3$ then value of the polynomial $p(-3)=-5$, where $p(x)=x^2+2x-8$.

This is the concise way to express the value of a polynomial.

Similarly, at $x=-4$, the value of the polynomial is,

$p(-4)=16-8-8=0$.

This value of $x$ is called a **ZERO** of the polynomial $p(x)=x^2+2x-8$. At this value of $x$, $p(x)=0$.

This is expressed as,

The

ZEROof a polynomial in $x$ is the value of $x$ at which the value of the polynomial is zero.

Further on, at $x=2$, the value of $p(x)=x^2+2x-8$ is,

$p(2)=2^2+4-8=0$.

This value of $x=2$ is the second Zero of the polynomial $p(x)=x^2+2x-8$.

Together, the two values of $x$ at which $p(x)$ is zero are called the **Zeroes of the polynomial.**

**As an alert**, let us consider the value of $p(0)$,

$p(0)=-8$.

This is *NOT a zero of the polynomial*. It represents just the coordinates $(\text{0, }-8)$ of the point of intersection of the polynomial curve with y-axis when the polynomial values are plotted on x-y coordinate system.

In general, a polynomial evaluates to $p(k)$ at $x=k$ with $k$ substituted for $x$ in the polynomial.

#### Number of Zeroes of a polynomial

A linear polynomial of degree 1 **has one Zero.**

A quadratic polynomial of degree 2 **may have AT MOST two numbers of Zeroes,** that is 0, 1 or 2 numbers of Zeroes.

A cubic polynomial of degree 3 **may have AT MOST three numbers of Zeroes,** that is, 1, 2 and 3 numbers of Zeroes (why not 0 number of Zeroes?).

In general, a polynomial $p(x)$ of degree $n$ may have at most $n$ numbers of Zeroes.

### Geometric representation of a polynomial and meaning of its Zeroes

You can draw a graph of a polynomial $p(x)$ on x-y coordinate axes by plotting the $x$ values on $x$ axis and corresponding polynomial values on $y$ axis. In concise form you can plot the graph for a polynomial as $y=p(x)$.

#### A linear polynomial plotted as a straight line on x-y coordinate axes

A linear polynomial will always be represented by a straight line that intersects the x-axis at a single point of zero of the polynomial. If a straight line is parallel to the x-axis without intersecting it, it will be represented by $y=b$, where $b$ is a constant. With coefficient of variable $x$ as zero, such a straight line can't be called a polynomial.

This is why *number of zeros of a linear polynomial is always 1.*

The following is the linear polynomial $p(x)=7x+3$ plotted on x-y coordinate axes.

To plot a straight line you need coordinates of just two points on the straight line. Joining the two you will get the straight line.

For the polynomial $y=7x+3$, at $x=-\displaystyle\frac{3}{7}$, $y=0$. The point $A\left(-\displaystyle\frac{3}{7},0\right)$ is the **zero of the polynomial.**

The second point is easy to obtain by substituting $x=0$ to get the value of $y$ as, $y=3$. This is the point $B$ with coordinates $B(0,3)$. Join the two points $A$ and $B$ to draw the straight line.

For a general linear polynomial $y=ax+b$, its zero is at the point $\left(-\displaystyle\frac{b}{a}, 0\right)$.

#### Geometric representation of a Quadratic polynomial and meaning of its zeros

Though you do not have to plot a quadratic polynomial perfectly on a x-y coordinate axes system, you can nevertheless get a rough plot of quadratic polynomial by finding the coordinates of a number of suitable points that satisfy the polynomial $y=p(x)$ and then joining the points.

Taking up plotting the graph for the quadratic polynomial $y=x^2+2x-8$, you can get the coordinates of the following points without much difficulty by trial and error on simple values of $x$.

$\text{A(-5, 7), B(3, 7)}$,

$\text{C(-4, 0), D(2, 0)}$,

$\text{E(-3, -5), F(1, -5)}$,

$\text{G(-2, -8), H(0, -8)}$, and,

$\text{I(-1, -9)}$.

**Verify** the values of $y$ by putting the values of $x$ in the polynomial $x^2+2x-8$ for each point coordinates.

**Join** the points and you will get a roughly drawn curve.

But in the following picture we show you the **actual plot** for the quadratic polynomial.

First important thing to know what the **Zeroes of the curve** for the quadratic polynomial $p(x)=x^2+2x-8$ are.

The Zeroes of the polynomial curve are simply the two points where the curve intersects the x-axis, points $C(-4, 0)$ and $D(2, 0)$.

This is true for any polynomial.

The points of intersection of the curve for a polynomial with x-axis are the Zeroes of the polynomial. At these values of $x$, the value of $y$, that is, the value of the polynomial becomes zero.

And from the picture you will actually get a visual feel of the nature of the curve for a polynomial as well its Zeroes.

**Algebraically**, you can find the Zeroes of the quadratic polynomial by **factorizing the following quadratic equation** and equating each factor to 0,

$y=x^2+2x-8=0$,

Or, $(x-2)(x+4)=0$.

The two roots of the equation are given by equating each factor to 0,

$(x-2)=0$,

Or, $x=2$, and,

$(x+4)=0$,

Or, $x=-4$.

Corresponding point coordinates of the Zeroes are, $(2, 0)$ and $(-4, 0)$.

The following section touches on a few additional aspects of the curve for a quadratic polynomial that you will learn later in details. If you wish, you may skip it.

#### Characteristics of the curve for a Quadratic polynomial

Observe that the curve for the quadratic polynomial is like a *gradually narrowing down well,* with a single point at the bottom at which value of $y$ reaches its minimum and the then rises with increase of $x$. This is called **minima of the quadratic polynomial.**

On both sides of the minima point $I(-1,-9)$, the two arms of the curve rise uniformly. This is actually a **parabolic curve** that you will learn later in details.

While rising, the two arms *cross the x-axis at the points that are the Zeroes of the polynomial* and then go on rising indefinitely.

Interestingly, the two rising arms of the curve on two sides of a line perpendicular to x-axis and passing through the minima are exact mirror images of each other.

This happens because any quadratic polynomial can be expressed as,

$(x+p)^2 \pm q^2$, where both $p$ and $q$ are real constants.

In fact, in this form you can easily determine the coordinates of the point at which the value of the polynomial becomes minimum (or maximum). The standard method of finding minima (or maxima) of a quadratic polynomial uses this form.

Now we'll see that the curve for all quadratic polynomials, though parabolic, won't be the same. For some, the curve will be like an inverted well having a maxima point rather than a minima; and for some, number of Zeroes won't be 2.

It may be 1 or even 0.

#### Geometric representation of a Quadratic polynomial with two Zeroes but having a point of maximum value of $y$

The following curve for the polynomial, $y=p(x)=-x^2+3x+4$ has two Zeroes at points $A(-1,0)$ and $B(4,0)$ and one point $C\left(\displaystyle\frac{3}{2}, \displaystyle\frac{25}{4}\right)$ at which value of $y$ reaches its maximum.

On both sides of this maxima, value of $y$ decreases indefinitely.

**Note:** You will notice that in this case, *the $x^2$ term is negative.* Whenever that happens, the quadratic polynomial would have an inverted well type curve with a maxima instead of a minima.

#### Geometric representation of a Quadratic polynomial with one zero

The following shows on the same plot, two curves for quadratic polynomials each with 1 zero, not 2.

The hallmark of such quadratic polynomials is,

The polynomial must be a square of sum of a linear polynomial.

For example, for $y=p(x)=x^2-4x+4$, the polynomial is actually,

$y=p(x)=(x-2)^2$.

Both Zeroes of $y$ have converged at $x=2$, that is, $B(2,0)$.

**Geometrically,** the curve just touches the x-axis at one point and does not cross it.

Similarly, in the second case,

$y=q(x)=-x^2+2x-1=-(x-1)^2$.

Both zeros converge at a single point $A(1,0)$ as the curve touches x-axis at this point.

#### Geometric representation of a Quadratic polynomial with NO ZERO

The following shows on the same plot, two curves for quadratic polynomials, **each with no Zero.**

On the same plot, the curves for the two types of quadratic polynomial (with minima and with maxima) are shown.

The polynomials are,

$y=p(x)=x^2-4x+5$, and,

$y=q(x)=-x^2+2x-2$.

None of the curves reach the x-axis for any value of $x$. The minima is above x-axis, and maxima below it.

**To sum up:**

Number of Zeroes of quadratic polynomials may be 2, 1 or 0, that is, maximum 2.

#### Mathematical significance of a quadratic polynomial having no Zeroes

In both the cases above, the curves for the quadratic polynomial do not touch or cross the x-axis resulting in no Zeroes of the polynomials.

This is because, in both cases, *the roots of the equation formed by equating the polynomial to 0 are imaginary.* In other words, the equivalent quadratic equations do not have any real valued roots that can be located on the real valued x-axis.

Whenever a maxima of a polynomial curve lies below the x-axis or the minima lies above the x-axis, it will indicate the presence of imaginary roots of the equivalent equation.

This is what happens in the case when number of Zeroes of a cubic polynomial is just 1 instead of 2 or 3. We'll deal with this case soon.

#### Geometric representation of a Cubic polynomial and its Zeroes

The following shows the geometric representation of a cubic polynomial, $y=p(x)=x^3+3x^2-4x-5$.

The curve plot for the cubic polynomial $y=p(x)=x^3+3x^2-4x-5$ has one maxima at point $A$ and a minima at point $B$.

On the left, with increasing $x$, value of $y$ rises, crosses the x-axis and after reaching the maxima starts to decrease. This decrease continues and the curve crosses the x-axis for the second time to reach its minima at $B$.

Henceforth with increasing $x$, value of $y$ goes on increasing, crosses the x-axis for the third time and continues increasing indefinitely.

All in all, **this cubic polynomial curve** has **three Zeros that are the intersection points of the curve with x-axis.**

This is the *maximum number of Zeros a cubic polynomial may have.*

The following shows the** second case of two Zeroes for a cubic polynomial.**

As you can see, the maxima $A$ is above the x-axis and contributes to one crossing, but the minima $B$ is right on the x-axis. The curve touches the x-axis at the minima and does not crosses it.

Only one Zero of the polynomial is contributed by this point $B$.

**Number of Zeroes reduces from 3 to 2.**

Because of the simple form of the cubic polynomial, $y=p(x)=x^3+4x^2$, we may examine the significance of its Zeroes from the resulting equation,

$y=x^3+4x^2=0$,

Or, $y=x^2(x+4)=0$.

Out of three roots of the cubic equation, two are converged at $x=0$, $y=0$, the origin, and the third is at,

$(x+4)=0$,

Or, $x=-4$.

From the curve plot, you should be able to see that indeed the curve crosses the x-axis at the point $C(-4,0)$ and touches it at the origin $B(0,0)$.

The following shows the third case of **one Zero of a cubic polynomial.**

In the third variation, *the maxima is above the x-axis as before and contributes to one Zero, but the minima is also above the x-axis** without touching or crossing it*.

**The number of Zeroes in this case is just 1.**

**Note:** This is the case of two imaginary roots with only one real valued root of the equivalent cubic equation.

**To sum up,**

Number of Zeroes of a cubic polynomial of degree 3 may be 3, 2 or 1, that is,AT MOST 3.

**In general,**

Number of Zeroes for a polynomial of degree $n$ is AT MOST $n$.

**Curiosity question:** Can you give an example of a cubic polynomial with NO ZEROES?

**Remember:** You do not have to bother about how to plot a quadratic or cubic polynomial, or what is the polynomial corresponding to a curve plotted on x-y axes. The main attention should be on the nature of the curve plot and its Zeroes.

### Solved problems: NCERT solutions for class 10 maths chapter 2 exercise 1: Geometrical meaning of Zeroes of Polynomials

#### Problem 1.i, Problem 1.ii.

The graphs of $y=p(x)$ for two polynomials $p(x)$ are shown below. Find the number of Zeroes of the polynomials.

#### Solution to problem 1.i.

The graph for polynomial $p(x)$ is a straight line parallel to the x-axis, that is, $y=p(x)=constant$. It will never cross the x-axis, and so the number of Zeroes is NIL.

**Answer:** 0.

#### Solution to problem 1.ii.

The graph for polynomial $p(x)$ is for a cubic polynomial that crosses the x-axis only once. So, the number of Zeroes is ONE.

**Answer:** 1.

#### Problem 1.iii, Problem 1.iv.

The graphs of $y=p(x)$ for two polynomials $p(x)$ are shown below. Find the number of Zeroes of the polynomials.

#### Solution to problem 1.iii.

The graph for polynomial $p(x)$ is for a cubic polynomial that crosses the x-axis three times. So, the number of Zeroes is THREE.

**Answer:** 3.

#### Solution to problem 1.iv.

The graph for polynomial $p(x)$ is for a quadratic polynomial that crosses the x-axis two times. So, the number of Zeroes is TWO.

**Answer:** 2.

#### Problem 1.v, Problem 1.vi.

The graphs of $y=p(x)$ for two polynomials $p(x)$ are shown below. Find the number of Zeroes of the polynomials.

#### Solution to problem 1.v.

The graph for polynomial $p(x)$ crosses the x-axis four times. So, the number of Zeroes is FOUR.

**Answer:** 4.

#### Solution to problem 1.vi.

The graph for polynomial $p(x)$ crosses the x-axis once and touches it two times. So, the number of Zeroes is THREE.

**Answer:** 3.

### End Note

This session involves graphs of polynomials that you don't have to draw. Main interest has been to know how the number of Zeroes of a polynomial behaves with change in shape of the polynomial curve.

Zeroes of a polynomial gains importance when it is converted to an equation by equating it to 0. The roots of the equation will be equivalent to the Zeroes of the polynomial equated to 0.

### NCERT Solutions for Class 10 Maths

#### Chapter 1: Real Numbers

**NCERT Solutions for Class 10 Maths on Real numbers part 1, Euclid’s division lemma puzzle solutions**

#### Chapter 2: Polynomials

**NCERT Solutions for Class 10 Maths Chaper 2 Polynomials 1 Geometrical Meaning of Zeroes of Polynomials**

#### Chapter 3: Linear Equations

**NCERT solutions for class 10 maths Chapter 3 Linear equations 7 Problem Collection**

**NCERT solutions for class 10 maths Chapter 3 Linear equations 6 Reducing non-linear to linear form**

**NCERT solutions for class 10 maths Chapter 3 Linear Equations 4 Algebraic solution by Elimination**

**NCERT solutions for class 10 maths Chapter 3 Linear Equations 3 Algebraic solution by Substitution**

**NCERT solutions for class 10 maths Chapter 3 Linear Equations 2 Graphical solutions**

**NCERT solutions for class 10 maths Chapter 3 Linear Equations 1 Graphical representation.**

#### Chapter 4: Quadratic equations

**NCERT solutions for class 10 maths Chapter 4 Quadratic Equations 1 What are quadratic equations**

**NCERT solutions for class 10 maths Chapter 4 Quadratic Equations 2 Solving by factorization**

**NCERT solutions for class 10 maths Chapter 4 Quadratic Equations 3 Solution by Completing the square**

#### Chapter 6: Triangles

**NCERT solutions for class 10 maths chapter 6 Triangles 1 Similarity of Triangles and Polygons**

**Solutions to Exercise 2 Chapter 6 NCERT X Maths, Characteristics of Similar triangles**

#### Chapter 8: Introduction to Trigonometry, Concepts and solutions to exercise problems

*NCERT solutions for class 10 maths chapter 8 Introduction to Trigonometry 1 Trigonometric Ratios*

*NCERT solutions for class 10 maths chapter 8 Introduction to Trigonometry 4 Trigonometric identities*

#### Chapter 8: Introduction to Trigonometry, only solutions to selected problems

**NCERT Solutions for Class 10 Maths on Trigonometry, solution set 6**

**NCERT Solutions for Class 10 Maths on Trigonometry, solution set 5**

**NCERT Solutions for Class 10 Maths on Trigonometry, solution set 4**

**NCERT Solutions for Class 10 Maths on Trigonometry, solution set 3**

**NCERT Solutions for Class 10 Maths on Trigonometry, solution set 2**

**NCERT Solutions for Class 10 Maths on Trigonometry, solution set 1**