Roots of Polynomials

Introduction

A wide group of algebraic expressions are combined to form the Polynomials. They can have constants, variables and exponents or, say, powers.

The powers of the variables are positive whole numbers and not any fractions when we consider any expressions of the polynomials.

Polynomials don't have any square root of variables or the negative powers on the variables.

The coefficient of a polynomial is the number multiplied by a variable.

The number which does not involve any variable or say, the number multiplied by the variable with power zero is called the constant of the polynomial.

The degree of the polynomial is the greatest power of any variable in a polynomial.

The terms of the polynomial are each part of the polynomial that is separated by addition or subtraction operations.

There is a wide range of real-life applications of Polynomials. In this tutorial, we will learn about polynomials, the roots of the polynomials and the factors of the polynomials.

Polynomials

Polynomial is the mathematical expression or, say, algebraic expression with one or more terms having non-zero coefficients. The leading term of the polynomial is the first term of the polynomial.

If the first term of the polynomial has the highest degree and the degree of the following terms are arranged in descending order of the exponents of the variables and the constant term, then the polynomial is called a Standard Polynomial.

The general form of Polynomial is given by,

$\mathrm{P(x)\:=\:\displaystyle\sum\limits_{j=0}^k\:\:b_{j}x^{j}\:=\:b_{0}x^{0}\:+\:b_{1}x^{1}\:+\:b_{2}x^{2}\:+\:...........\:+\:b_{k\:-\:2}x^{k\:-\:2}\:+\:b_{k\:-\:1}x^{k\:-\:1}\:+\:b_{k}x^{k}}$

Polynomials are classified into two types according to their degree and number of terms.

According to the degree of the polynomial, they are majorly classified into seven types, namely, Sextic Polynomial, Quintic Polynomial, Quartic Polynomial, Cubic Polynomial, Quadratic Polynomial, Linear Polynomial, Zero polynomial and Constant Polynomial.

Type of Polynomial	Polynomial Degree	Example
Sextic Polynomial	6	$\mathrm{5x^{6}\:+\:3x^{5}\:-\:x^{4}\:-\:7}$
Quintic Polynomial	5	$\mathrm{x^{5}\:-\:2x^{4}\:+\:x^{3}\:+\:9x^{2}\:-\:32}$
Quartic Polynomial	4	$\mathrm{42x^{4}\:+\:23x^{3}\:+\:12x^{2}\:-\:11x\:-\:20}$
Cubic Polynomial	3	$\mathrm{9x^{3}\:+\:15x^{2}\:+\:8x\:-\:8}$
Quadratic Polynomial	2	$\mathrm{7x^{2}\:-\:x\:-\:21}$
Linear Polynomial	1	$\mathrm{6x\:+\:42}$
Zero polynomial	0	0
Constant Polynomial	0	$\mathrm{23x^{0}}$

According to the number of terms in a polynomial, they are majorly classified into four types, namely, quadrinomial, trinomial, binomial and monomial.

Type of Polynomial	Number of terms	Example
Quadrinomial	4	$\mathrm{7x^{3}\:+\:x^{2}\:+\:x\:-\:5}$
Trinomial	3	$\mathrm{6x^{4}\:+\:12x\:+\:78}$
Binomial	2	$\mathrm{4x^{2}\:-\:120}$
Monomial	1	$\mathrm{45x\:(or)\:15x\:+\:30x}$

Roots of Polynomials

The solutions of the polynomials 𝑃(𝑥) are called the roots or the zeroes of the polynomials.

In the polynomial $\mathrm{p(x)\:=\:\displaystyle\sum\limits_{j=0}^k\:\:b_{j}x^{j}}$ the roots of the polynomial 𝑃(𝑥) are simply the values of the variable 𝑏 for which the Polynomial

In simple terms, If 𝑏 is the root of the polynomial 𝑃(𝑥), then 𝑃(𝑏) is equal to zero. The formula to find the root of a linear polynomial $\mathrm{P(x)\:=\:bx\:+\:d}$ is given by,

$$\mathrm{x\:=\:-\:\frac{d}{b}}$$

The formula for finding the roots of a quadratic polynomial $\mathrm{P(x)\:=\:bx^{2}\:+\:cx\:+\:d}$ is given by,

$$\mathrm{x\:=\:\frac{-c\:\pm \sqrt{c^{2}\:-\:4bd}}{2b}}$$

Thus, the root of the polynomial 𝑥 can be found with the help of the formula.

Factors of Polynomials

The root of the polynomial is written in the form of a polynomial with degree one is called the factor of the polynomial.

If 4 is the root of the polynomial 𝑃(𝑥), then the factor of 𝑃(𝑥) is written as $\mathrm{x\:+\:4}$

It is said that the degree of a polynomial and the number of factors of the polynomial will be equal.

Solved Examples

1. The Polynomial $\mathrm{P(x)\:=\:x^{2}\:+\:12x\:+\:32}$, find the roots of 𝑃(𝑥).

Solution −

Put $\mathrm{P(x)\:=\:0}$

$$\mathrm{\Longrightarrow\:x^{2}\:+\:12x\:+\:32\:=\:0}$$

$$\mathrm{x\:=\:\frac{-c\:\pm \sqrt{c^{2}\:-\:4bd}}{2b}}$$

$$\mathrm{x\:=\:\frac{-12\:\pm \sqrt{(12)^{2}\:-\:4(1)(32)}}{2(1)}}$$

$$\mathrm{x\:=\:\frac{-12\:\pm \sqrt{144\:-\:128}}{2}}$$

$$\mathrm{x\:=\:\frac{-12\:\pm \:4}{2}}$$

$$\mathrm{x\:=\:\frac{-12\:+\:4}{2}\:,\:\frac{-12\:-\:4}{2}}$$

$$\mathrm{\frac{-8}{2}\:,\:\frac{-16}{2}}$$

$$\mathrm{x\:-4\:,\:-8}$$

The values of 𝑥 are −4 and −8.

Thus, the roots of the given polynomial $\mathrm{P(x)\:are\:-4\:and\:-8}$

2. If a given polynomial is of degree two, $\mathrm{P(x)\:=\:x^{2}\:+\:7x\:+\:10}$ Find the factors of $\mathrm{P(x)}$

Solution −

Put 𝑃(𝑥) = 0

$$\mathrm{\Longrightarrow\:x^{2}\:+\:7x\:+\:10\:=\:0}$$

$$\mathrm{\mathrm{x^{2}+\:5x\:+\:2x\:10\:=\:0}}$$

$$\mathrm{x(x\:+\:5)\:+\:2(x\:+\:5)\:=\:0}$$

$$\mathrm{(x\:+\:5)(x\:+\:2)\:=\:0}$$

Therefore the factors of the given polynomial $\mathrm{P(x)\:are\:(x\:+\:5)\:and\:(x\:+\:2)\:=\:0}$

In the given polynomial, the degree of the polynomial and the number of the polynomial are 2.

3. The Polynomial $\mathrm{P(x)\:=\:x^{2}\:+\:16x\:+\:63\:=\:0}$ find the roots of 𝑃(𝑥).

Solution −

Put 𝑃(𝑥) = 0

$$\mathrm{\Longrightarrow\:x^{2}\:+\:16x\:+\:63\:=\:0}$$

$$\mathrm{x\:=\:\frac{-c\:\pm \sqrt{c^{2}\:-\:4bd}}{2b}}$$

$$\mathrm{x\:=\:\frac{-16\:\pm \sqrt{(16)^{2}\:-\:4(1)(63)}}{2(1)}}$$

$$\mathrm{x\:=\:\frac{-16\:\pm \sqrt{256\:-\:252}}{2}}$$

$$\mathrm{x\:=\:\frac{-16\:\pm \sqrt{4}}{2}}$$

$$\mathrm{x\:=\:\frac{-16\:\pm \:2}{2}}$$

$$\mathrm{x\:=\:\frac{-16\:+\:2}{2}\:,\:\frac{-16\:-\:2}{2}}$$

$$\mathrm{x\:=\:\frac{-14}{2}\:,\:\frac{-18}{2}}$$

$$\mathrm{x\:=\:-7\:,\:-9}$$

The values of 𝑥 are −7 and −9

Thus, the roots of the given polynomial 𝑃(𝑥) are −7 and −9.

4. If the Polynomial $\mathrm{22x\:-\:44}$ find the roots of 𝑃(𝑥).

Solution −

Put 𝑃(𝑥) = 0

$$\mathrm{22x\:-\:44\:=\:0}$$

$$\mathrm{22x\:=\:44}$$

$$\mathrm{x\:=\:\frac{44}{22}}$$

𝑥 = 22 is the root of the given polynomial

5. The Polynomial $\mathrm{P(x)\:=\:x^{2}\:+\:15x\:+\:54}$ Find the factors of 𝑃(𝑥).

Solution −

Put 𝑃(𝑥) = 0

$$\mathrm{\Longrightarrow\:x^{2}\:+\:15x\:+\:54\:=\:0}$$

$$\mathrm{x^{2}\:+\:9x\:+\:6x\:+\:54\:=\:0}$$

$$\mathrm{x(x\:+\:9)\:+\:6(x\:+\:9)\:=\:0}$$

$$\mathrm{(x\:+\:9)(x\:+\:6)\:=\:0}$$

Therefore the factors of the given polynomia $\mathrm{P(x)\:are\:(x\:+\:9)\:and\:(x\:+\:6)}$

6. Identify the types of the given polynomials −

• $\mathrm{3x^{5}\:-\:66}$

• $\mathrm{7x^{6}\:+\:12x^{3}\:+\:8x^{2}\:-\:54}$

• $\mathrm{x^{2}\:-\:11x\:-\:30}$

• $\mathrm{32x\:+\:4}$

Solution −

A	$\mathrm{3x^{5}\:-\:66}$	Quintic Polynomial
B	$\mathrm{7x^{6}\:+\:12x^{3}\:+\:8x^{2}\:-\:54}$	Sextic Polynomial
C	$\mathrm{x^{2}\:-\:11x\:-\:30}$	Quadratic Polynomial
D	$\mathrm{32x\:+\:4}$	Linear Polynomial

Conclusion

• Polynomial is the algebraic expression with one or more terms having nonzero coefficients.

• The root of the polynomial 𝑃(𝑥) is b, then 𝑃(𝑏) is equal to zero.

• Solutions to the polynomials 𝑃(𝑥) are called the roots or the zeroes of the polynomials.

• The degree of the polynomial is the greatest power of any variable in a polynomial.

• The degree of a polynomial and the number of factors of the polynomial will be equal.

FAQs

1. Give some real-life applications of Polynomials?

• Polynomials are used in the construction of buildings

• They are widely used in safety measures such as predicting traffic patterns both on the internet and road.

• The growth of certain germs can be found with the help of polynomials.

• They can be applied in economic production.

• Their contributions include in stock-exchange, marketing and finance.

2. What are the applications of polynomials in science?

• To find the population growth of both humans and animals.

• To discover the birth and death rates of animals.

• To calculate the number of chopped down trees in a forest and balance it by replantation

• To manage the soil for agricultural purposes.

• To determine certain molecular compositions.

3. Who is called the father of Polynomials?

The Greek mathematician Diophantus of Alexandria is called the father of polynomials. He is the author of a series of books called Arithmetica. His works dealt with solving algebraic equations. Some of his contributions are Diophantine equations and Diophantine approximations.

4. What are algebraic expressions?

The expressions with variables and constants along with mathematical operations are called algebraic expressions. It can also be said that they are formed by integer variables and constants. The algebraic terms build the algebraic expressions.

E.g $\mathrm{3x^{2}\:-\:7x\:+\:9}$

5. What is meant by factorisation of the polynomial?

If a polynomial is broken down into the product of its factors or say product of smaller polynomials, then the process is called factorisation of the polynomial. Through factorisation of the polynomial, we find the factors of the polynomial.