Polynomials: Definitions and Operations

In algebra, polynomials are fundamental algebraic expressions that support a vast array of mathematical concepts and applications.

In this article, we will discuss the world of polynomials, exploring their structure, operations, and practical applications.

What Are Polynomials?

Polynomial is an expression consisting of variables and coefficients. It involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

For example, x, 5x^2 – 3x + 7, and 2xy^3 + 4yz are all polynomials.

General Form of a Polynomial

The general form of a polynomial is:

f(x) = an * x^n + an-1 * x^(n-1) + ... + a2 * x^2 + a1 * x + a0

Here, f(x) represents the polynomial, x is the variable, and an, an-1, ..., a0 are the coefficients. The exponents n should be non-negative integers.

Degree of a Polynomial

Degree of a polynomial is determined by the term with the highest exponent.

For example, the polynomial 5x^3 + 2x^2 – x + 7 has a degree of 3, since the highest exponent is on the x^3 term.

Degree gives information about the highest power that variable has in the polynomial. A constant polynomial, like 5, has a degree of 0.

Polynomial vs. Non-Polynomial

Polynomials are distinct from non-polynomial expressions due to their structure. For an expression to be considered a polynomial,

• Variables must have non-negative integer exponents.
• Coefficients should be real numbers.
• There must be a finite number of terms.
• Only addition, subtraction, and multiplication operations are allowed.

Polynomial Types

Constant Polynomials (Degree 0)

These have no variable terms. For example, P(x) = 7.

Linear Polynomials (Degree 1)

These have a single variable term. For example, Q(x) = 3x + 2.

Quadratic Polynomials (Degree 2)

These have two variable terms. For example, R(x) = 2x^2 - 5x + 1.

Cubic Polynomials (Degree 3)

These have three variable terms. For example, S(x) = 4x^3 + x^2 - 3x - 7.

Quartic Polynomials (Degree 4)

These have four variable terms. For example, T(x) = 4x^4 + 3x^3 - x^2 + x - 7

Polynomial Type	Degree	Example
Zero Polynomial	Not Defined	P(x) = 0
Constant Polynomial	0	Q(x) = 8
Linear Polynomial	1	R(x) = 2x + 5
Quadratic Polynomial	2	S(x) = 3x^2 + 2x + 1
Cubic Polynomial	3	T(x) = 4x^3 + 2x^2 + x + 7
Quartic Polynomial	4	U(x) = 5x^4 + 4x^3 + 2x^2 + 3x + 1

Polynomial Operations

Adding and Subtracting Polynomials

To add or subtract polynomials, combine like terms. Like terms have the same variables raised to the same powers.

Example

(3x^2 – 2x + 5) + (2x^2 + 4x – 1) = 5x^2 + 2x + 4

To subtract polynomials, change the sign of the polynomial being subtracted and then combine like terms.

Example

(3x^2 + 5x – 7) – (2x^2 – 4x + 1) = x^2 + 9x – 8

Multiplying Polynomials

To multiply two polynomials together, use the distributive property and FOIL method. Multiply each term in one polynomial by each term in the other.

Example

(x + 3)(x – 5)

= x(x) – x(5) + 3(x) – 3(5)

= x^2 – 5x + 3x – 15

= x^2 – 2x – 15

The degree of the product is the sum of the degrees of the polynomials being multiplied.

Dividing Polynomials

Long division can be used to divide polynomials. Divide the first term of the dividend by the first term of the divisor.

Example

(x^3 + 3x^2 – 9) / (x – 3)

The result is x^2 + 6x + 9.

Special Polynomials

Monomials

These are polynomials with a single term. For example, 5x or 4y^3.

Binomials

These are polynomials with two terms. For example, 3x - 2 or a^2 + b.

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