Remainder Theorem & Polynomials

Introduction

The remainder theorem is used to find the remainder when a polynomial is divided by another polynomial. Polynomials are algebraic expressions consisting of different algebraic terms & these terms are joined together by mathematical operators like addition (+) & subtraction(-).

The concept of polynomials is used in almost every field of mathematics. Also, the polynomial is considered one of the important branches of calculus. It also has wide applications in science. It is a central concept of algebra & algebraic geometry. It is used to form polynomial equations & word problems for analysing & solve difficult problems. In this tutorial, we are going to study polynomials, different types of polynomials & theorems used to solve polynomials. i.e remainder theorem & factor theorem.

What are polynomials?

The word polynomials are made up of two words ‘poly’ & ‘nominal’. ‘Poly’ means & many means ‘nominal’ term. It is an expression consisting of two or more algebraic terms having different power & same variable, these terms are joined together by a mathematical operator. Terms in the polynomials consist of variables, constants, coefficients & mathematical operators.

• Variable − It is an alphabet used to represent unknown values. These values are varying.

• Coefficient − A number which is multiple of a variable.

• Constants − It is a number having a fixed value. It doesn’t change in any mathematical condition.

• Mathematical operator − These symbols are used to represent and perform basic arithmetical operations like addition, subtraction, multiplication & division.

• Exponent − It is the power to be raised of any variable.

For example, $\mathrm{5x^{2}\:+\:2x\:-\:3}$

Here 𝑥 is a variable , 5 & 2 are the coefficient of $\mathrm{x^{2}\:and\:x\:respectively}$, 3 is constant value + & − are mathematical operator & 2 is an exponent of variable 𝑥.

Polynomials are classified on two basis −

• Classification of polynomial based on the number of terms

• Classification of polynomial based on the degree

Classification of polynomial based on the number of terms

According to the number of terms polynomials are classified into 3 types −

i) Monomials

An algebraic expression consists of only one term is known as monomials

For example, $\mathrm{5x\:,\:\frac{4}{3}m}$

ii) Binomials

An algebraic expression consists of two terms known as binomials.

For example, $\mathrm{2x\:-\:3x\:,\:5m\:-\:3m}$

iii) Trinomials

An algebraic expression consisting of three terms is known as the trinomials.

For example, $\mathrm{8x^{2}\:+\:5x^{2}\:-\:2x\:,\:5x^{2}\:+\:3x\:-\:2}$

Classification of polynomial based on the degree

Furthur moving to the classification of polynomial based on the degree first we need to understand the concept of degree. The degree of any polynomial is the highest power of the variable

According to the degree of variables polynomials in one variable are classified into three types −

i) Linear polynomial

Polynomial with degree 1 is known as the linear polynomial.

The standard form of a linear polynomial is $\mathrm{ax\:+\:b}$ here, 𝑎 & 𝑏 are real numbers and a is not equal to 0.

For example, $\mathrm{3x\:-\:1\:,\:7m}$

ii) Quadratic polynomials

Polynomial with degree 2 are known as the quadratic polynomial

A standard form of a quadratic polynomial is $\mathrm{ax^{2}\:+\:bx\:c}$

Where 𝑎, 𝑏 & 𝑐 are real numbers and a is not equal to 0

For example, $\mathrm{2x^{2}\:-\:3x\:+\:5\:,\:-2y^{2}}$

iii) Cubic polynomial

Polynomial with degree 3 is known as the cubic polynomial.

A standard form of a cubic polynomial is $\mathrm{ax^{3}\:+\:bx^{2}\:+\:cx\:+\:d}$

Where $\mathrm{a\:,\:b\:,\:c\:\&\:d}$ are real numbers and a is not equal to 0.

For example, $\mathrm{x^{3}\:+\:3x^{2}\:+\:5x\:+\:2}$

Polynomials can be denoted as $\mathrm{p(x)\:,\:q(m)\:,\:r(y)}$ according to which type of variable is present in the polynomial. These polynomials can be written in three forms first is a standard form, the second is the coefficient form & other is the index form.

For example, write $\mathrm{3m^{5}\:-\:7m\:+\:5m^{3}\:+\:2\:\:\:\&\:\:\:x^{4}\:-\:3x^{3}\:+\:2x^{2}\:+\:5x\:-\:7}$ in standard, coefficient & index form

Polynomial in standard form	Polynomial in coefficient form	Polynomial in index form
$\mathrm{3m^{5}\:+\:5m^{3}\:-\:7m\:+\:2}$	$\mathrm{(3\:,\:0\:,\:5\:,\:0\:,\:-7\:,\:2)}$	$\mathrm{3m^{5}\:+\:0m^{4}\:+\:5m^{3}\:-\:0m^{2}\:-\:7m\:+\:2}$
$\mathrm{x^{4}\:-\:3x^{3}\:+\:5x\:+\:2x^{2}\:-\:7}$	$\mathrm{(1\:,\:-3\:,\:2\:,\:5\:,\:,\:-7)}$	$\mathrm{x^{4}\:-\:3x^{3}\:+\:2x^{2}\:-\:5x\:-\:7}$

What is factor theorem?

Factor theorem is used to find the roots & factors of the polynomial. This theorem links factors & zeros of the polynomial.

Before moving to the factor theorem first we need to understand the term zeros of a polynomial

Zeros of a polynomial

Zeros of the polynomial are the points where the value of the polynomial becomes zero.

Factor theorem: It 𝑝(𝑥) is a polynomial of degree n & 𝑎 is any real number the

• $\mathrm{(x\:-\:a)\:is\:a\:factor\:of\:p(x)\:,\:if\:p(a)\:=\:0\:\&}$

• $\mathrm{(a)\:=\:0\:,\:if\:(x\:-\:a)\:is\:factor\:of\:p{x}}$

Proof − $\mathrm{Dividend\:=\:Divisor\:\times\:Quotient\:+\:Remainder}$

$$\mathrm{p(x)\:=\:(x\:-\:a)\:\times\:q(x)}$$

By using remainder theorem,

$$\mathrm{p(x)\:=\:(x\:-\:a)\times\:q(x)\:+\:p(a)}$$

Putting 𝑝(𝑎) = 0 we get,

$$\mathrm{p(x)\:=\:(x\:-\:a)\times\:q(x)\:+\:0}$$

Therefore, $\mathrm{p(x)\:=\:(x\:-\:a)\times\:q(x)}$

So we can say that $\mathrm{(x\:-\:a)}$ is a factor of the polynomial 𝑝(𝑥).

Steps to find factors of a polynomial by using factor theorem

• Divide the given polynomial 𝑝(𝑥) by given $\mathrm{(x\:-\:a)}$

• After division, confirm whether the remainder is zero or not. If the remainder is not zero, it indicates that $\mathrm{(x\:-\:a)}$ is not a factor of 𝑝(𝑥).

• By using division, write the given polynomial as the product of $\mathrm{(x\:-\:a)}$& the quadratic quotient.

• Express the given polynomial as the product of its factors.

What is the remainder theorem?

This theorem is used to find the remainder when a polynomial is divided by a linear polynomial. When we performed division, the leftover number or term is known as the remainder. So let's discuss the remainder theorem.

Remainder theorem

Let $\mathrm{p(x)}$ be any polynomial having a degree greater than or equal to 1 & let x be any real number. Suppose $\mathrm{p(x)}$ is divided by $\mathrm{x\:-\:a}$ , quotient $\mathrm{q(x)}$ & the remainder is $\mathrm{r(x)}$ , then the remainder is 𝑝(𝑎).

Proof − Let $\mathrm{p(x)}$ be any polynomial having degree greater than or equal to 1 & let be any real number. Suppose $\mathrm{p(x)}$ is divided by $\mathrm{x\:-\:a}$ , quotient $\mathrm{q(x)}$ & the remainder is $\mathrm{r(x)}$. Mathematically this can be represented as

$$\mathrm{p(x)\:=\:(x\:-\:a)\times\:q(x)\:+\:r(x)}$$

Here the degree of $\mathrm{x\:-\:a}$ is one & degree of 𝑟(𝑥) less than the degree of $\mathrm{x\:-\:a}$

Therefore degree of 𝑟(𝑥) = 0. It means 𝑟(𝑥) is constant, say r

$$\mathrm{p(x)\:=\:(x\:-\:a)\times\:q(x)\:+\:r}$$

In particular, if we consider $\mathrm{x\:=\:a}$ , this equation will give us,

$$\mathrm{p(a)\:=\:(a\:-\:a)\:\times\:q(a)\:+\:r}$$

Hence proved.

Divisibility & factors of polynomials by remainder theorem:

Divisibility of polynomials:

Consider three polynomials $\mathrm{p(x)\:,\:q(x)\:\&\:r(x)}$

Therefore $\mathrm{p(x)\:=\:q(x)\:.\:r(x)}$

If $\mathrm{p(x)\:,\:q(x)\:\&\:r(x)}$ are polynomials having integer coefficients, then we can say that $\mathrm{p(x)}$ is divisible by $\mathrm{q(x)}$.

This property is also applicable to rational numbers & complex coefficients

Factors of a polynomial:

If $\mathrm{p(x)\:=\:q(x)\:.\:r(x)}$ then we can say that , $\mathrm{q(x)}$& $\mathrm{r(x)}$ are factors of $\mathrm{p(x)}$.

Solved examples

1) By using the remainder theorem divide $\mathrm{x^{4}\:-\:5x^{2}\:-\:4x\:by\:x\:+\:3}$ & find the remainder.

Answer − Here, dividend polynomial $\mathrm{p(x)\:=\:x^{4}\:-\:5x^{2}\:-\:4x}$

$\mathrm{Divisor\:=\:x\:+\:3}$

$\mathrm{Take\:x\:=\:3}$

Putting the value of 𝑥 in dividend polynomial it will give,

$\mathrm{p(3)\:=\:(-3)^{4}\:-\:-\:5(-3)^{2}\:-\:4(-3)}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:=\:81\:-\:45\:+\:12}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:=\:48}$

Therefore remainder is 48

2) If the polynomial $\mathrm{t^{3}\:-\:3t^{2}\:+\:kt\:+\:50}$ is divided by −𝟑 , the remainder is 62 then find the value of 𝒌.

Answer − When given polynomial $\mathrm{t^{3}\:-\:3t^{2}\:+\:kt\:+\:50}$ is divided by $\mathrm{t\:-\:3}$ then the remainder is 62 it means the value of the polynomial when 𝑡 = 3 is 62.

Therefore $\mathrm{p(t)\:=\:t^{3}\:-\:3t^{2}\:+\:kt\:+\:50}$

By using remainder theorem,

$$\mathrm{p(3)\:=\:3^{3}\:-\:3\times\:3^{2}\:+\:k\times\:3\:+\:50}$$

$$\mathrm{\:=\:27\:-\:27\:+\:3k\:+\:50}$$

$$\mathrm{=\:3k\:+\:50}$$

But the remainder is $\mathrm{62\:.............(Given)}$

Therefore

$\mathrm{3k\:+\:50\:=\:62}$

$\mathrm{3k\:=\:62\:-\:50}$

$\mathrm{3k\:=\:12}$

$\mathrm{k\:=\:4}$

3) If $\mathrm{p(x)\:=\:x^{3}\:+\:4x\:-\:5}$ divided by $\mathrm{x\:-\:1}$ then find the remainder & also check whether $\mathrm{x\:-\:1}$ is a factor of 𝒑(𝒙) or not ?

Answer − Here $\mathrm{p(x)\:x^{3}\:+\:4x\:-\:5}$

$\mathrm{p(1)\:=\:\:1^{3}\:+\:4\times\:1\:-\:5}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:=\:1\:+\:4\:-\:5}$

Here remainder is zero,

Therefore, according to remainder theorem

$\mathrm{x\:-\:1}$ is a factor of 𝑝(𝑥) .

4) Factorise $\mathrm{(x\:+\:2)\:(x\:-\:3)\:(x\:-\:7)\:(x\:-\:2)\:+\:64}$

Answer − $\mathrm{(x\:+\:2)\:(x\:-\:3)\:(x\:-\:7)\:(x\:-\:2)\:+\:64}$

$\mathrm{\:\:\:\:\:=\:(x^{2}\:-\:5x\:-\:4)(x^{2}\:-\:5x\:+\:6)\:+\:64}$

Putting $\mathrm{x^{2}\:-\:5x\:=\:m}$

$\mathrm{=\:(m\:-\:14)\:(m\:+\:6)\:+\:63}$

$\mathrm{=\:m^{2}\:-\:14m\:+\:6m\:-\:84\:+\:84}$

$\mathrm{=\:m^{2}\:-\:8m\:-\:20}$

$\mathrm{=\:(m\:-\:10)\:(m\:+\:2)}$

$\mathrm{=\:(x^{2}\:-\:5x\:-\:10)\:(x^{2}\:-\:5x\:+\:2)\:.............(replacing\:x^{2}\:-\:5x\:with\:m)}$

Conclusion

This tutorial covers the topic of remainder theorem & polynomials. We have learned about polynomials, different types of polynomials, different forms of polynomials, remainder theorem & factor theorem, along with examples.

Polynomials are the algebraic expressions obtained from algebraic terms. This is a central concept in algebra & algebraic geometry. It has wide applications in the field of science & mathematics. In geometry, it is used to represent the perimeter & area of a shape as well as the volume of a solid. It is also used to represent weather patterns in meteorology. This tutorial will surely help you to understand the remainder theorem & polynomials.

FAQs

1. What is quadrinomial?

Quadrinomial is a type of polynomial having four terms.

For example, $\mathrm{m^{4}\:-\:2m^{2}\:-\:5\:=\:0}$

2. What is the difference between remainder theorem & factor theorem?

The remainder theorem links the remainder of division by a binomial with the value of a function at a point, whereas the factor theorem links the factor of a polynomial to its zero.

3. State one application of the remainder theorem?

The main application of the remainder theorem is the factor theorem. The factor theorem is derived from the remainder theorem & is used to determine the roots of the polynomial.

4. State the applications of the factor theorem?

In real life, the concept of factoring can be used when exchanging money, dividing any quantity into equal parts & comparing prices.

5. State the importance of the factor theorem?

It is a special type of polynomial theorem used to find the roots or factors of the polynomial. This is the easiest method to find a factor of polynomials.