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A polynomial equation is one of the foundational concepts of algebra in mathematics. Having a clear and logical sense of how to solve a polynomial problem will allow students to be much more efficient in their examinations and will also act as a firm base in their higher studies. Polynomial equations are in the forms of numbers and variables. A polynomial equation is a form of an algebraic equation. There is a minute difference between a polynomial and polynomial equation. Polynomials are expressions whereas polynomial equations are expressions equated to zero.

We will try to understand polynomial equations in detail. We will learn about the degree of a polynomial, types of a polynomial equation and most importantly, how to solve a polynomial equation.

A polynomial equation is an expression consisting of variables, coefficients and exponents. A polynomial function is one which has a single independent variable. The exponents in a polynomial equation can only be in the form of positive integers, therefore, any negative integer exponent disqualifies as a polynomial equation.

The independent variable can occur multiple times in a polynomial. These different occurrences of the variable are separated by operations of addition, subtraction and multiplication. The degree of the polynomial is defined as the highest degree of exponent that exists in the equation. It is also called the order of the polynomial equation.

For the polynomial x^{2} + 3x + 6 , the degree or the order of the polynomial is 2.

A polynomial is generally of the form \[a_{n}x^{n}\]. Here, a is called the coefficient, x is the independent variable and n is the exponent. Equating this polynomial to zero gives us a polynomial equation. The value of the exponent n can only be a positive integer as discussed above.

Any polynomial function can be of the form,

\[F(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + . . . + a_{1}x + a_{1} = 0\] is the general formula of a polynomial.

On putting the values of a and n, we will obtain a polynomial function of degree n.

\[F(x) = 2x^{2} + 5x = 0\]

Here, the polynomial 2x^{2} + 5x, equated to zero gives us the polynomial equation F(x) = 2x^{2 }+ 5x = 0 with degree 2.

Polynomial equations are classified upon the degree of the polynomial. For practical reasons, we distinguish polynomial equations into four types.

Monomial/Linear Equation

A polynomial equation with only one variable term is called a monomial equation. It is also called a linear equation. The algebraic form of a linear equation is of the form:

ax + b=0, where a is the coefficient, b is the constant and the degree of the polynomial is 1.

Examples:

2x + 10 = 0

x - 5 = 0

Binomial/Quadratic Equation

A polynomial with two variable terms is called a binomial equation. It can also be called a quadratic equation. The algebraic form of a quadratic equation is of the form:

ax^{2} + bx + c = 0, where a and b are coefficients, c is the constant and degree of the polynomial is 2.

Examples:

2x^{2} + 2x + 2 = 0

x^{2} - 4=0

Trinomial/Cubic Equation

A polynomial with three variable terms is called a trinomial equation. It is also called a cubic equation. The algebraic form of a quadratic equation is of the form:

ax^{3} + bx^{2} + cx + d = 0, where a, b and c are coefficients, d is the constant and degree of the polynomial is 3.

Examples:

x^{3} + 2x^{2} + 3x - 5 = 0

2x^{3} - 5x = 0

Polynomial Equation

A polynomial with more than three variable terms is called a polynomial equation. It is of the form \[a_{n}x^{n} + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + . . . + a_{1}x + a_{1} = 0\].

Examples,

4x^{4 }+ 2x^{3 }+ x^{2 }+ 5 = 0

10x^{5} + 2x - 10 = 0

Polynomial equations are generally solved with the hit and trial method. We put in the value of the independent variable and try to get the value of expression equal to zero.

In case of a linear equation, obtaining the value of the independent variable is simple. We solve the equation for the value of zero.

For the polynomial, 2x - 4 = 0

2x = 4

x = 2

However, this solution is not easily applicable in higher degrees of polynomials, therefore, we go with the hit and trial method.

FAQ (Frequently Asked Questions)

Q1. How do we Solve a Quadratic Polynomial Formula?

Ans: One method to solve a quadratic formula is to use the hit and trial method, where we put in different values for the independent variable and try to get the value of the expression equal to zero. However, this method of hit and trial can be tiresome, so we try to find the roots of the equation using the quadratic formula.

x = [-b ± √(b^{2} - 4ac)]/2a

A quadratic equation is of the form of ax^{2} + bx + c = 0, where a and b are coefficients and the degree of the equation is 2, which means that there are two roots of the equation

x = [-b ± √(b^{2} - 4ac)]/2

Using the quadratic formula, we obtain the roots of the equations instantly.

Q2. Solve the Following Polynomial Equation, 5x^{2} + 6x + 1 = 0.

Ans: The above equation is a polynomial equation with degree 2. It is a quadratic equation with two roots.

The equation 5x^{2 }+ 6x + 1 = 0 is a quadratic equation, where a,b and c are real numbers.

a = 5

b = 6

c = 1

So, using the quadratic formula,

x = [-b ± √(b^{2} - 4ac)]/2

x = [-6 ± √(6^{2} - 4 x 5 x 1)]/2

x = [-6 ± √(36 - 20)]/2

x = [-6 ± √(16)]/2

x = [-6 ± 4]/2

The value of x are,

x = -2/2 = -1

x = -10/2 = -5

The roots of the equation are -1 and -5. So the values of x that satisfy the equation are -1 and -5.