Polynomial Equations

Polynomial equations are those expressions which are made up of multiple constants and variables. The standard form of writing a polynomial equation is to put the highest degree first and then, at last, the constant term. An example of a polynomial equation is:

0 = a⁴ +3a³ -2a² +a +1

Polynomial Functions

A polynomial function is an expression constructed with one or more terms of variables with constant exponents. If there are real numbers denoted by a, then function with one variable and of degree n can be written as:

Solving Polynomials

Any polynomial can be easily solved using basic algebra and factorization concepts. While solving the polynomial equation, the first step is to set the right-hand side as 0. The explanation of a polynomial solution is explained in two different ways:

• Solving Linear Polynomials
• Solving Quadratic Polynomials

Solving Linear Polynomials

Getting the solution of linear polynomials is easy and simple. First, isolate the variable term and make the equation as equal to zero. Then solve as basic algebra operation. An example of finding the solution of a linear equation is given below:

Example: Solve 3x – 9

Solution:

First, make the equation as 0. So,

3x – 9 = 0

⇒ 3x = 9

⇒ x = 9/3

Or, x = 3.

Thus, the solution of 3x-9 is x = 3.

Solving Quadratic Polynomials

To solve a quadratic polynomial, first, rewrite the expression in the descending order of degree. Then, equate the equation and perform polynomial factorization to get the solution of the equation. An example to find the solution of a quadratic polynomial is given below for better understanding.

Example: Solve 3x² – 6x + x³ – 18

Solution:

First, arrange the polynomial in the descending order of degree and equate to zero.

⇒ x³ + 3x² -6x – 18 = 0

Now, take the common terms.

x²(x+3) – 6(x+3) =0

⇒ (x²-6)(x+3)=0

So, the solutions will be x =-3 or

x² = 6

Or, x = ±√6