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# Polynomials: Types and Examples

## What is a Polynomial?

A polynomial is a mathematical equation that consists of variables and coefficients. It involves only the operations of addition, subtraction, and multiplication. The variables in a polynomial can have non-negative integer exponents.
For example, the expression 3x2 – 2x + 5 is a polynomial.
where,

• 3, -2, and 5 are coefficients,
• x is a variable,
• and 2 and 1 are the exponents.

The highest exponent in this polynomial is 2, so it is called a quadratic polynomial. Polynomials can have any degree, which is the highest exponent of the variable in the polynomial.

### Examples:

(i)  3x + 5 is a polynomial in x of degree 1.

(ii) 4y2 -3y +8 is a polynomial in x of degree 2.

(iii) 2u3 + 4u2 -5u + √3 is a polynomial in u of degree 3.

(iv)√(x+3), 1/(x+2), etc, are not polynomials

## Polynomial or Not?

These are polynomials:

• 5x
• x − 4
• −4y2 − (79)x
• 4xyz + 3xy2z − 0.1xz − 200y + 0.5
• 512v5 + 99w5
• 9

(Yes, “9” is a polynomial because it can be just a constant!)

These are not polynomials

• 5xy-4 is not polynomials, because the exponent is “-4” (exponents can only be allowed 0,1,2,…)
• 2/(x+4) is not, because dividing by a variable is not allowed
• 1/x is not either
• √x is not, because the exponent is “½”.

## Degree of a Polynomial

The degree of a polynomial is the highest power (or exponent) of the variable in the polynomial equation.
For example, in the polynomial expression 4u4 – 5u3+ 6u2 -8u+3, the degree of the polynomial is 4, which is the highest power of the variable u. This is a fourth-degree polynomial.
Here are given below some properties described to the degree of a polynomial:

• The degree of a non-zero constant polynomial is 0.
• The degree of a non-zero linear polynomial in one variable is 1.
• The degree of the sum or difference of two polynomials is the highest degree of the two polynomials.
• The degree of the product of two polynomials is the sum of the degrees of the two polynomials.

Here are some examples of polynomials of different degrees:

• Degree 0: Constant polynomial (e.g. 6)
• Degree 1: Linear polynomial (e.g. 5x – 6)
• Degree 2: Quadratic polynomial (e.g. x2 + 2x + 1)
• Degree 3: Cubic polynomial (e.g. 2x3 – 5x2 + 3x + 7)
• Degree n: nth degree polynomial (e.g. 4xn + 2xn-1 + 5xn-2 } – 3)
Polynomial Degree Example
Constant or Zero Polynomial 0 10
Linear Polynomial 1 5x+3
Cubic Polynomial 3 2u3-3u3+8u+1
Quartic Polynomial 4 2y4+3y3-5y2+9y+1

### Related Topics

Example: Find the degree of the polynomial 4u4 – 5u3+ 6u2 -8u+3

Solution:

The degree of the polynomial is 4.

## Operations on Polynomials

Several operations can be performed on polynomials. These operations are the basis for solving polynomials.
Here are given some of the most common operations on polynomials:

Polynomials can be added and subtracted by combining like terms.

For example, (3x2 + 2x + 1) + (2x2 – 3x + 2) = 5x2 – x + 3.

#### 2. Multiplication

Polynomials can be multiplied using the distributive property.

For example, (x + 2)(x – 3) = x2 – x – 6.

#### 3. Division

Polynomials can be divided using long division or synthetic division.

For example, (x2 + 2x + 1) ÷ (x + 1) = x + 1.

#### 4. Evaluation

Polynomials can be evaluated for a specific value of the variable.

For example, P(x) = x2 + 2x + 1, P(2) = 9.

#### 5. Factoring

Polynomials can be factored into their constituent factors.

For example, x2 + 3x + 2 can be factored as (x + 1)(x + 2).

#### 6. Integration

The integral of a polynomial can be found using the power rule.

For example, if P(x) = x3 + 2x2 + x, then the indefinite integral of P(x) with respect to x is (1/4)x4 + (2/3)x3 + (1/2)x2 + C, where C is a constant.

#### 7. Derivatives:

The derivative of a polynomial can be found using the power rule.

For example, if P(x) = x3 + 2x2 + x, then P'(x) = 3x2 + 4x + 1.

## Types of Polynomials

Polynomials are of three separate types and are classified based on the number of terms in it. The three types of polynomials are given below:

1. Monomial
2. Binomial
3. Trinomial
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