How to deal with Polynomials

In mathematics general equations are in the form of any polynomial equation which includes multiple terms in it. Polynomial expressions are formed with constants and variables with product of integer coefficient or without integer coefficient. These variables and constants are related to each other by arithmetic operators to form the polynomial equation. Any polynomial equation consists of finite number of derivatives. The standard representation of polynomial equation is in the form of following mathematical expression:

p_nxⁿ+.....+ p₂x²+ p₁x¹+ p₀x⁰,

here 'n' is a finite number which tells the highest no. of derivatives in any polynomial equation and ( pn, pn-1, ….... p0) are integer coefficients. Polynomial equation's derivatives may have various order of degree. If all the derivatives are of same order than that expression is said to be as linear polynomial equations otherwise non-linear polynomial equations.

Let us take an example of polynomial equations to explore it more

x²+ 5x + 3 = 4 -------> (1)

y =2 ---------> (2)

x³+ y³=1 -------> (3)

Every polynomial equation is said to be of n- order polynomial equation and the value of 'n' is the highest degree among all the derivatives in the equation.

So in the above examples, the equation first is said to be as 2 – order non -linear polynomial equation because it has highest degree as 2 in all its derivatives and all the derivatives are not of same order that's why it is in non -linear form.

Polynomial functions are also called as monomials, binomials and trinomials depending upon the number of unknown variables in the polynomial function.

If the polynomial function consist derivatives of a single unknown variable than it is said to be a monomial and if derivatives includes two unknown variables than it will be a binomial and so on trimonial on presence of 3 unknown variables.

In the above examples equation (1) and (3) are binomials and (2) is of monomial form

How to Solve Polynomials

Polynomial expressions are the most common way of representing mathematical equations with multiple terns as the name polynomial suggests , Poly = “multiple” and nomial = ”terms ” Polynomial expressions includes variables with or without integer coefficient and constants related to each other by normal arithmetic operators to form an equation. The standard form of any polynomial equation is as:

cn xⁿ+.....+ c2 x²+ c1x¹+ c0x⁰,

here cn , …....c2, c1,c0 are constant coefficient terms and x is variable with n to 0 order of degree.

Polynomial functions consist of finite number of derivatives in it. Let's take an example of polynomial equations for better understanding:

5x²+ 2x + 3 = 4

x =2

x²+ x³=1

Every polynomial equation is called as n- order polynomial equation and the value of 'n' is the highest degree among all the derivatives in the equation.

So the equation first is said to be as 2 – order polynomial equation.

Polynomial is further described according to the present number of variables in the equation. Any polynomial equation is said to be monomial if it consist only one variable in it .

^x2+ 5x =3

here 'x' is the only variable in the equation so above one is a monomial equation.

If polynomial equation consists of 2 variables than it is said to be as binomial and if 3 than as trinomial.

Like x²– y²= 3 ( a binomial equation or a equation with two monomials )

x³- 2y + z = 5 ( a trinomial with three variables (x, y, z))

While performing multiplication between two polynomial terms, it just get complicated. In that case you only have to sum the products of each term multiplied of first polynomial by each term of second polynomial as in the example below:

(x²+ y²) ( x + y) =3

x³+x²y +y²x + y³=3

it gives 3 order polynomial equation as x³+ y³+ xy²+ x²y = 3

Polynomial Functions, Binomial, Trinomial

A polynomial is a mathematical expression that involves a sum of powers in one or more variables multiplied by the co-efficients. When we define the polynomial in one variable it is called as the univariate polynomial with constant coefficients and it is given by the formula:

a_nxⁿ+.....+ a₂x²+ a₁x¹+ a₀x⁰,

Monomials contains the individual summands with the coefficients. Here, a_n≠ 0. n is the degree of the x and a is the constant.

Polynomial functions may contain multiple powers of x, but its degree is calculated on the basis of highest power. Let's take an example to understand how to calculate the degree of polynomial functions.

5x²+ 7x³+ 8x+ 1,

in this expression, we are having three values of 'x', that is x , x²and x³.

In this case the degree of x = 3 as the highest power of x in the expression is three.

Do you know what are binomial functions? No, relax i will tell you, binomial are those polynomial functions that contain two variables or we can say it contains two terms . The binomial x²– y² can be factored as the product of two other binomials that is:

x²– y²= (x+y)*(x -y). The product of binomial terms is called as binomial product.

For example: x+5, 2y+7, etc.

Now, lets talk about trinomial it is a polynomial function that consists of the three terms or monomials.

That means it includes the three variables. Any expression containing three terms is defined as the trinomial expression. For illustration: 4x + 6y -3z it Is a trinomial with three variables x, y, z. Do you know what is trinomial equation, it involves three terms. For example x = y+zm is trinomail equation. you can learn more about all the topics using online help.

 

 

How to use polynomial functions

A polynomial function is mathematical expression which involves a sum of powers in one or more variables are multiplied by coefficients. Polynomial Functions are easy to understand, low degree Polynomial equations can be simplified explicitly. A polynomial function in independent variable 'y' is function of f(y), which is given be a formula:

anyn+......+ a2y2+ a1y1+a0.

Here, an,a2,a1, a0 are the real numbers, and these are the coefficients of the variable y. We assume that an≠ 0. The number n shows the degree of the polynomial function.

The individual summands with the coefficients generally included are called as monomials whereas the product in multivariate case i.e. with the coefficients omitted are called terms. Let's take any polynomial equation

3x + 1=0,

This polynomial equation has degree 1, as the highest power of x is 1. Since the degree of equation is 1, it is a linear equation. In the same way the degree of Polynomial equations are determined.

Do you know what are Quadratic equations? Quadratic equations are defined as the polynomial equation having degree as 2, means power of x is 2. Quadratic equations are of form ax2+ bx +c = 0, where, x is a variable and a,b,c are constants and the value of a ≠ 0. For solving Quadratic equations we mainly use two methods one is factorizing and other is quadratic equation formula. In the first method we try to find factors and the quadratic formula is given as:-

using this formula we can solve almost every quadratic equation.

In algebra we spend lots of time in solving Polynomial equations or factoring polynomials. So you must be strong in solving Polynomial equations. For this, you can take online help. Online help is easy to use and it offers lots of services to you.