Standard Form of Polynomial

A polynomial is said to be in its standard form when it is written in decreasing order of power. In standard form, the highest degree variable comes first in the polynomial, followed by the next terms in decreasing order of power. If the terms of the polynomial are arranged in a decreasing order of power, then it is known as the standard form of a polynomial.

The standard form of the polynomial is f(n) = a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + … +a₁x + a₀, where x is the variable and a_i are the coefficients.

For example, if the given equation is 2x + 3x² - 5, then the standard form of the polynomial is 3x² + 2x - 5. Here, the highest degree 3x² is written first, followed by the next highest term, 2x, and finally the constant term, 5.

The highest power of the variable in the polynomial defines the degree of the polynomial. In 3x² + 2x³, the degree of the polynomial is 3 as it is the highest power. The degree of the polynomial can be determined in two ways, they are:

• Degree of a single variable polynomial
   
• Degree of a multivariable polynomial

The highest exponent or the highest power in the given polynomial expression is known as the degree of the single variable polynomial. It is determined by the term with the highest exponent of the polynomial. The polynomial with a single term consists of only one term, which may include a variable and a coefficient. The degree of the constant polynomial is always zero, as there is no variable. 

A multivariable polynomial contains more than one variable, such as x, y, z, etc. For finding the degree of a multivariable polynomial, we need to add the exponents of the variables in each term, and the highest such sum of exponents per term of the polynomial is considered the degree of the polynomial. To find the degree of the polynomial in 5x³y² - 3xy³ + 2x, we need to add the exponents of all the variables in each term. 

The degree of 5x³y² is 3 + 2 = 5.

The degree of 3xy³ is 1 + 3 = 4.

The degree of 2x is 1.

Therefore, the degree of the polynomial is 5 as it is the highest power.

Polynomials can be classified into four types based on the number of terms and the highest degree. Classification of polynomials is as follows:

• Monomial
   
• Binomial
   
• Trinomial
   
• Multinomial

Monomial: A monomial is a polynomial that consists of only one term. Example: 3x or 2x².

Binomial: A polynomial that has two terms is called a binomial. Some examples of binomials are 2x² + 3 or 3xy + 2x.

Trinomial: A polynomial with three terms is known as a trinomial. 2x² - 3x +8 is a trinomial polynomial.

Multinomial: A polynomial with more than three terms is called a multinomial. For example, 3x³ + 2x² - 3x + 4.

The polynomials which are classified based on degree are,

• Constant Polynomial
   
• Linear Polynomial
   
• Quadratic Polynomial
   
• Cubic Polynomial

Constant Polynomial: A polynomial with a degree of 0 is called a constant polynomial. It does not have any variables. Constant polynomials are -2 or 5.

Linear Polynomial: The polynomial with a degree of 1 is called a linear polynomial. When we graph a linear polynomial, it will be in a straight line. 5x + 3 is a linear polynomial.

Quadratic Polynomial: A polynomial with a degree two is called a quadratic polynomial. For example, 3x² + 2x is a quadratic polynomial.

Cubic Polynomial: A polynomial of degree three is a cubic polynomial. 3x³ + 2x² + x is an example of a cubic polynomial.

Adding and subtracting are basic operations used to combine polynomials by working with like terms. Adding and subtracting polynomials is similar to adding and subtracting numbers. We add numbers by using their place values; likewise, polynomials are added by like terms. Like terms are the terms that have the same variable and the same powers. Once we match the like terms, we can add or subtract their numbers or coefficients. We can see how to add and subtract polynomials using simple examples.

Add: (3x + 2) + (5x + 4)

To add the given polynomials, we need to add the like terms together from both polynomials. 

Here, the terms are 3x and 5x, then 2 and 4. 

3x + 5x = 8x

2 + 4 = 6

After adding the like terms together, we will get the new polynomial as 8x + 6.

The real-life applications of the standard form of polynomials help to understand the use of polynomials in daily life. Some real-life applications of the standard form of polynomials are,

• Engineering: Engineers use polynomials to design the shapes of arches and bridges. Polynomials help them by telling them how high the arch is at different points.

• Sports: In basketball or soccer, polynomials are used to track how the ball moves in the air. This helps the coach and the players to understand how high and far the ball will go.

• Agriculture: Farmers use polynomials to predict crop yields using fertilizers. It also helps them to choose the right amount of fertilizers.

• Computer Graphics: Polynomials help in creating curves and shapes in video games. Designers use polynomials to create smooth animations.