How to tackle 8th grade Polynomial Functions

In my opinion, Polynomial functions are one of the interesting areas of studies but sometimes it comes with so much complexities that it becomes quite a bit difficult to solve them. So the most necessary thing to understand is that we need to practice it a lot and it requires full concentration while solving it. Polynomial is basically a term which deals in almost every type of mathematical equations or statements. The most common terminologies used in polynomial expressions comes in eighth standard mathematics are monomial, binomial and trinomials. Algebraic equation with all variables having whole number, exponents or powers are called polynomials. The expressions in which the power of variables are negative and which include rational numbers are not polynomials. Algebraic expression having single term is known as Monomial and expression with two terms are known as Binomial whereas expressions with more than two terms or having three terms are known as Trinomials.

Now lets talk about Polynomial Functions. A polynomial functions p is basically a function or an expression that can be formed by combining the variable and some constants by a finite number of additions, subtractions, and multiplications.

A polynomial equation comes with the sum of the power of same derivatives and includes different integer constants, while the derivatives used are finite in numbers. The standard form of any polynomial equation is as:

b_n yⁿ + b_n-1 y^n-1 + ….......................+. b₂ y² + ….... + b₀ y⁰

Lets take some examples of Polynomial Equations to understand it better.

Function p (x) = x² – 3 is a polynomial of degree 2. in standard form it can be represented as : a₂ = 1, a₁= 0 and a₀ = -3.

An example to show a non polynomial function is P (d) = 1/d. This function becomes arbitrarily large for values of d close to zero and no polynomial does that.

Some examples of polynomial equations

A polynomial function f, is a function of the form f(x) = anxn + an-1xn-1 + ... + a2x2 + a1x1 + a0 where a0, a1...an are real numbers. It will soon become more understandable  with some more examples. If "n" is not zero, then f is said to have degree n. A polynomial f(x) with real coefficients and of degree n has n zeros (not necessarily all different). Some or all are real zeros and appear as x-intercepts when f(x) is graphed. To make it more simple lets start by explaining the word polynomial, it is the word which comes made of two terms "poly" states many and nominal states "terms". Nomenclature of different  polynomial functions, depending upon the terms present in the equation is done such as, if it has only one term it is called as monomial, if two terms it is called as binomial and if three terms it is called as trinomial, and so on with increasing variable terms.

Lets make it more elaborate with the help of examples, X + X>2 = 4  is an example of monomial, x + y = 5 is an example of binomial, X + Y + Z = 7 is an example of trinomial. When there are equations involving polynomial it is known as a Polynomial equation. For solution of a polynomial equation different values for variables in the equation satisfying the equation along with the given constant coefficient values used in the polynomial.

Lets see it practically with the help of some example, 5 x + 6 y = 0 is a polynomial equation, for a point A in a plane having coordinates (0, 0). Co-ordinates states value of x = 0, and y = 0 for this equation. Substituting values of x and y in the Polynomial equations  we have 5 (0) + 6 (0) = 0 + 0 = 0. As the values on both sides of the equation are equal this is a solution of the equation. Hence point A is the solution for this Polynomial equations.

TutorVista help on Polynomials

Polynomial is one of the most important term used in mathematical world which plays an important role in almost every type of mathematical equations or statements. Terminologies or concepts used in Polynomial equations are are monomial, binomial and trinomials. Algebraic equation with all variables having whole number, exponents or powers are called polynomials. Monomials are the Algebraic expression consist of single term and those algebraic expression comes equipped with two terms are known as Binomial Whereas the expressions with more than two terms or having three terms are known as Trinomials.


Lets talk about Polynomial functions used in mathematics. Polynomial function includes various things like terms, factors, variables, and constants. Let us talk about all the above terminologies in detail which are required to form a polynomial function. Terms can be explained as when numbers are implemented with addition or subtraction are known as terms. Terms can be further divided in to two sections that are Like terms or Unlike terms. Terms that has the same power of the same variables are called Like terms. The terms used in an expression that do not contain the same power of the same variables are called unlike terms. In an expression if the product of the numbers are used then the expression is called as factors. Variables are just representing a symbol which uses different values under it whereas constant is a single value symbol.


A polynomial equations comes with the sum of the power of same derivatives and includes different integer constants, while the derivatives used are finite in numbers.


Lets take some examples of Polynomial equations to understand it better.


7xyz : Monomial


x + 7y : Binomial


x + 3y – c : Trinomial


 

Learn Polynomial functions by taking help of Tutorvista

In mathematics, problems are represented in expression form on which standard principles of math are applicable. The major part of mathematical problems is represented in form of any polynomial function. A polynomial is a mathematical expression which may consist of various derivatives of various orders which are related with each other by arithmetical operators to form an math expression.

The standard form of any polynomial is as:

c_nyⁿ + c_n-1 y^n-1 + .......................+c₂ y² + ….... + c₀y⁰

The standard form consists of derivatives of only one variable, but as the name suggests “poly” means many ,  and “nomial” means terms, it means polynomial may consist of derivatives of different variables in it. According to this property polynomial equations are further categorizes as: monomial, binomial, trinomial and so on. Every polynomial function consist of different terms, factors, variables and constant integers coefficients. If two numbers or variables are combined through addition or subtraction operator then they form a term and if the degree of all those terms is same then they are said to be like terms otherwise unlike terms. The numbers are multiplied or product of two numbers is used in a polynomial than that part of it is known as factor. Variable is a symbol which represents the value which may change and constant is a fixed numerical value.

Let's see few examples of polynomial equations to explore more about it:

y – y² = 2 (a Monomial equation)

x + y = 2 (Binomial equation)

x + y - z = 1 (Trinomial equation)

All the algebraic equations like linear equations, quadratic equations, etc and they are different kind of polynomial equations also and for solving these polynomial equations the variables of the functions should be replaced by appropriate numerical value which will satisfy the equation in the end. To learn more about this topic switch to tutorvista.

Solving polynomial equations of mathematics

Polynomial is basically a term which deals in almost every type of mathematical equations or statements. The most common terminologies used in polynomial expressions are monomial, binomial and trinomials. Algebraic equation with all variables having whole number, exponents or powers is called polynomial. The expressions in which the power of variables are negative and rational numbers are not polynomials. Algebraic expression having single term known as monomial and expression with two terms are known as binomial whereas expressions with more than two terms or having three terms are known as trinomials.

Polynomial functions come equipped with terms, factors, variables, and constants. Let us explore about all these required objects to form a polynomial function. When numbers are implemented with addition or subtraction then they are said to be terms.Terms are of two types Like terms or Unlike terms. Terms that have the same power of the same variable are called like terms. The terms used in an expression that do not contain the same power of the same variables are called unlike terms. In an expression if the product of the numbers are used then the expression is called as factors. Variables are just representing a symbol which uses different values under it whereas constant is a single value symbol.

Polynomial equations, come with the sum of the power of same derivatives and includes different integer constants, while the derivatives used are finite in numbers. The standard form of any polynomial equation is as:

b_nyⁿ + b_n-1 yⁿ-1 + ….......................+. b₂ y² + ….... + b₀y⁰

In the above equation, y is the variable and b is the integer coefficient used.

Lets see some of the examples of polynomial equations to understand it better.

10xyz : Monomial

3x + 7y : Binomial

3x + 7y – c : Trinomial

How to Solve Polynomials

Polynomial is a term which implies with every kind of mathematical expression. Polynomial function consists of terms, factors, variables, and constants. Let us explore about all these required objects to form a polynomial function, When numbers are implemented with addition or subtraction than they are said to be terms, when product of the numbers are used than that form is called factors. Variables are just representing a symbol which uses different values under it whereas constant is a single value symbol.

A polynomial equation includes sum of the power of same derivatives with different integer coefficients and these all derivatives are finite in numbers. The standard form of any polynomial equation is as:

b_n yⁿ + b_n-1 y^n-1 + ….......................+. b₂ y² + ….... + b₀ y⁰

Here y is the variable with n types of derivatives, 'b' is an integer co-efficient and 'n' represents the finite number of derivatives in polynomial equation.

Sometimes any polynomial equation may consist of number of different derivatives or variables. So according to this property polynomial is categorized in various types which are binomial, monomial, trinomial and so on.

A polynomial is said to be a monomial when it only have one single variable derivatives and if equations consist of derivatives of two variables than that is a form of binomial equation. Similarly a trinomial will include derivatives of 3 different variables. Let us take some examples of polynomial equations:

x – x² = 2 (a Monomial)

x + y = 1 (Binomial)

x + y + z = 3 (Trinomial)

In mathematics most of the equations are in form of polynomial equation like every algebraic equation is a type of polynomial equation. So it is clear that polynomial functions are important in mathematics equation formation so for enhancing your knowledge in this topic and various other math topics you can take online math help on a math tutoring website “ TutorVista”.

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.