What is a Polynomial?

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 a_n, 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.