Solving Polynomials

What are Polynomials?
The simple definition of Polynomial is that, it is an expression containing multiple terms that are combined together through addition, subtraction and multiplication. This polynomial definition is illustrated by this polynomial equation: 3x + 4y + 5. In this example, the polynomial equation is a combination of three terms.

Ways of Solving Polynomials
There are different ways of solving polynomials and the strategy differs based on the polynomial equation provided. In this article, let us discuss two different ways to solve polynomial through relevant examples.

Example 1: Solve polynomial (a+8) (2a-10)

In this equation, there are two sets of terms that are multiplied with each other. To solve this polynomial, multiply each term of the first set with all the terms in the second set. This is illustrated below:

When ‘a’ is multiplied with ‘2a’, it gives 2a2. Then ‘a’ is multiplied with -10, which results in ‘-10a’. As a next step, multiply 8 with the terms in the second set. When ‘+8’ is multiplied with ‘2a’, it gives ‘+16a’ and ‘+8’ multiplied with ‘-10’ gives ‘-80’. Now the equation is dissolved into:

(a+8) (2a-10) = 2a2 -10a + 16a -80

In the above equation, the second term (-10a) and the third term (+16a) contains the same term with the same degree, but the constant value alone varies. When there are several terms with same variable and same degree, they can be combined. In this case, -10a+16a can be rearranged as 16a-10a, which when solved results in 6a. Thus, the polynomial equation is further solved into the result 2a2 + 6a – 80.

Example 2: Solve polynomial equation a+2b = 5a+7b
In this equation, there are two polynomials, one on the left hand side and the other on the right hand side. In this case, these polynomials help each other to find the solution. The polynomial solver for this expression will be evolved through the steps below:
• Original Expression: a+2b = 5a+7b
• Moving right hand side expression to left: a +2b-5a-7b=0.
• Grouping terms of same variables and degree: a-5a+2b-7b=0
• Combining terms of same variables and degree: -4a-5b
• Take common factor out: -1(4a+5b)

Thus the result is -1(4a+5b).

When the polynomials are complex, the above strategies alone might not work. For instance, if you are solving quadratic polynomials then you have to perform series of steps such as finding zeros of the polynomial, finding roots and much more to solve them.