Polynomial Addition Algorithm

Addition of two polynomials involves combining like terms present in the two polynomials.


By like terms we mean the terms having same variable and same exponent.
For example two terms are like only if:

  •  The two terms have same variable
  •  The two terms have same power of the variable

e.g.` 3x^2` and `5x^2 `are like terms. Similarly` 4x^2y` and ` 2x^2y ` are like terms.
However `3x^2y` and `xy^2 `are not like terms, since one involves `x^2y` and the other has `xy^2`. Hence the powers of the variables involved are not the same.

Before we proceed towards more complex examples of polynomials addition, let us solve a few simple examples that make the idea of like terms clear.


Polynomial Addition Examples


Example 1:
  `6x`

Explanation:

First we need to observe whether the given terms are like or not.

Both 4x and 2x involve a single variable x, each having the same exponent (=1). Hence the two terms can be simply added as indicated by the plus (+) sign.

=> `4x + 2x = (4 + 2)x = 6x`


Example 2:
 ` xy + 4xy^2`

Explanation:

First we need to observe whether the given terms are like or not.

We notice that the first two terms, that is 3xy and 2xy are like, since both the terms involve same variables x and y with the exponents of both the variables being 1. Hence we can subtract 2xy from 3xy as mentioned by the -ve sign.

However, the third term that is 4xy2 is not like the first two terms and therefore cannot be combined with them through addition.

=> `3xy-2xy + 4xy^2 = (3-2)xy + 4xy^2 `

=> ` 3xy-2xy + 4xy^2 ` = ` xy + 4xy^2` (Cannot be simplified further)


Example 3:
 ` 5x^3 + 3x^2 + 10x^2y + 2xy + 4xy^2 - 6y^2 - 5`

Explanation:

We combine the like terms while leaving the un-like terms as they are. Hence

` (5x^3 + 3x^2y + 4xy - 6y^2) + (3x^2 + 7x^2y - 2xy + 4xy^2 - 5) `

= ` 5x^3 + 3x^2 + (3 + 7)x^2y + (4 - 2)xy + 4xy^2 - 6y^2 - 5 `

= ` 5x^3 + 3x^2 + 10x^2y + 2xy + 4xy^2 - 6y^2 - 5`

is the required result.


Example 4:
 ` 8x + 4`

Explanation:

Given above is a simple algebraic expression that involves summation of two polynomials. There's nothing to be confused about the parentheses, they are simply showing addition / subtraction.

We simply combine the like terms:

`[(3x - 5) + 2x] - [(6x - 2) - (9x + 7)] `

` = [3x - 5 + 2x] - [6x - 2 - 9x - 7] `

` = [3x + 2x - 5] - [6x - 9x - 2 - 7] `

` = [(3 + 2)x - 5 ] - [(6 - 9)x - (2 + 7)] `

` = [5x - 5] - [- 3x - 9] `

` = 5x - 5 + 3x + 9 `

` = 5x + 3x - 5 + 9 `

` = 8x + 4`

= 8

8x + 4 is the simplified result for the given expression.

Sometimes it may be easier to add two polynomials by placing the like terms above and below each other. This method is known as "vertical method" as opposed to the "horizontal method" discussed above. We present one example below that will help you understand this technique of polynomial addition:


Example 5:
 ` 5x^3 + 3x^2 + 10x^2y + 2xy + 4xy^2 - 6y^2 - 5`

Explanation:

` 5x^3 + 3x^2y + 4xy - 6y^2 `

` 3x^2 + 7x^2y - 2xy + 4xy^2 - 5`

+

_______________________________________________

` 5x^3 + 3x^2 + 10x^2y + 2xy + 4xy^2 - 6y^2 - 5 `

Which has the same result as we obtained in Example 3 above.






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