﻿ Polynomials - Definition, Types, Questions, Remainder Theorem

# What is a Polynomial?

## Polynomials - Sub Topics

• What is a Polynomial?
• General Form of a Polynomial
• Types of Polynomials
• Polynomials in One Variable
• Remainder Theorem of Polynomial
• Factor Theorem of Polynomials
• Addition and Subtraction of Polynomials
• The reading material provided on this page for Polynomials is specifically designed for students in grades 9 to 12. So, let's begin!

## What is a Polynomial?

A polynomial is a mathematical expression that consists of variables and coefficients, which are combined using addition, subtraction and multiplication operations. The variables in a polynomial have exponents that are whole numbers.

For example: 3x2 + 2x – 1, 4x3 + 2x2 - 5x + 1, 2x4 + 3x3 - x2 + 5x – 2, 5x + 1, x2, etc.

## General Form of a Polynomial

A polynomial is a mathematical expression that is made up of one or more terms, each of which contains a variable raised to a non-negative integer power.

where P(x) is polynomial, x is variable and an, a(n-1), ..., a1, a0 are the coefficients of the terms. The exponent n is known as the degree of the polynomial and it represents the highest power of x in the polynomial.

Example: P(x) = 3x2 + 2x - 5
This is a polynomial of degree 2, with coefficients of 3, 2 and -5. The terms are 3x2, 2x, and -5, respectively.

## Types of Polynomials

There are two types of variables based on degree and terms.

### Types of Polynomial Based on Terms

1. Monomial: A polynomial with only one term, such as x3, 7x2 and so on.
2. Binomial: A polynomial with two terms, such as x2 + 3x, 5x – 2 and so on.
3. Trinomial: A polynomial with three terms, such as x3 + 2x2 - 4x, 3x2 - 5x + 2 and so on.
4. Quadrinomial: A polynomial with four terms, such as x4 - 31x3 + 8x2 - 4, 3x5 - 5x2 + 3x - 7 and so on.

### Types of Polynomial Based on Degree

1. Monomial: A polynomial with a degree of 2, such as 7x +1, x/5 and so on.
2. Binomial: A polynomial with a degree of 2, such as x2 + 2x + 1, 3x2 - 4x – 5 and so on.
3. Cubic: A polynomial with a degree of 3, such as x3 + 3x2 + 3x + 1, 2x3 – 5x2 + 3x – 4 and so on.
4. Quartic: A polynomial with a degree of 4, such as x4 + 2x3 – 5x2 + 4x + 1, 3x4 – 2x + 1 and so on.
5. Quintic: A polynomial with a degree of 5, such as x5 + 2x4 - 3x3 - 5x + 1, 3x5 - 5x2 + 2x – 1 and so on.

## Polynomials in One Variable

Polynomials in one variable are mathematical expressions that consist of a sum of powers of a single variable, usually represented by x, multiplied by constant coefficients.

For example, a polynomial in one variable could be written as:

3x2 + 4x - 5

In this example, x is the variable and 3, 4, and -5 are the constant coefficients. The exponent on x (2 and 1) indicates the degree of the polynomial. In this case, the degree of the polynomial is 2 because the highest exponent on x is 2.

## Remainder Theorem of Polynomial

The remainder theorem states that if a polynomial P(x) is divided by (x-a), the remainder is equal to P(a).

For example, let's say we have the polynomial P(x) = x3 + 2x2 + 3x + 4 and we want to divide it by (x – 2).

Using long division, we get:

=(x3 + 2x2 + 3x + 4) ÷ (x – 2)

=x2 + 4x + 11, with a remainder of 26

By remainder theorem -

P(2) = 2³ + 2 x 2² + 3 x 2 + 4

= 8 + 8 + 6 + 4

= 26

So, P(2) = 26, which is the remainder of the division.

This confirms that the remainder theorem holds for this example.

## Factor Theorem of Polynomials

It states that if a polynomial f(x) evaluates to zero when a particular value (say "a") is substituted for x, then (x - a) is a factor of f(x) and the converse of it is also true.
Mathematically, the Factor Theorem can be expressed as follows:

(i) If f(a) = 0, then (x - a) is a factor of f(x).
(ii) If (x – a) is a factor of f(x), then f(a) = 0.

For example, let's say we have a polynomial f(x) = x2 - 4x + 4. If we substitute x = 2 into the polynomial, we get f(2) = (2)2 - 4(2) + 4 = 4 - 8 + 4 = 0.

Since f(2) = 0, we can conclude that (x - 2) is a factor of f(x).

## Addition and Subtraction of Polynomials

To add and subtract polynomials, you follow similar rules as you do with numbers. Here are the general steps to add and subtract polynomials:

i. Align the like terms.

ii. Add or subtract the coefficients.

iii. Combine like terms.

Here's an example that demonstrates adding and subtracting polynomials:

Example 1: Add the polynomials:

2x2 + 5x + 3 and -x2 + 2x - 1

= 2x2 + 5x + 3 + (-x2 + 2x - 1)

= 2x² + 5x + 3 - x² - 2x - 1

= 2x² - x² + 5x – 2x + 3 -1

= x² + 3x + 2

So, the sum of the given polynomials is x2 + 3x + 2.

Example 2: Subtract the polynomials: 4x3 - 3x2 + 2x - 5 and 2x3 + x2 + 3x + 1

= 4x3 - 3x2 + 2x – 5 - (2x3 + x2 + 3x + 1)

= 4x3 - 3x2 + 2x – 5 - 2x3 - x2 - 3x – 1

= 4x3 - 2x3 - 3x2 - x2 + 2x - 3x – 5 – 1

= 2x³ - 4x² - x - 6

Therefore, the difference between the given polynomials is 2x3 - 4x2 - x - 6.

Remember to align like terms, perform addition or subtraction of coefficients, and simplify the resulting polynomial by combining like terms.

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