Adding and Subtracting Polynomials

Method

Adding and Subtracting Polynomials

Given two polynomials, their sum or difference can be calculated. Consider the following pair of polynomials. - 2x^2 + x + 3 and 2x^2 + 4x - 10 In order to add these polynomials, there are three steps to follow. Note that subtraction of the polynomials can be performed by applying the same three steps, only instead of adding the like terms, they will be subtracted.

1
Rearrange Terms
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Generally speaking, polynomials are added by adding their like terms. Therefore, it can be helpful to rearrange the terms so that the like terms are next to each other.

(- 2x^2 + x + 3) + (2x^2 + 4x - 10)
- 2x^2 + x + 3 + 2x^2 + 4x - 10
(- 2x^2 + 2x^2) + (x + 4x) + (3 - 10)

When subtracting polynomials, remember to distribute - 1 to the terms of the second polynomial before rearranging the terms.

(- 2x^2 + x + 3) - (2x^2 + 4x - 10)
- 2x^2 + x + 3 - (2x^2 + 4x - 10)
- 2x^2 + x + 3 - 2x^2 - 4x + 10
(- 2x^2 - 2x^2) + (x - 4x) + (3 + 10)

2
Add and Subtract Like Terms
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Next, the like terms can be combined. In other words, the monomials should be added or subtracted by adding or subtracting their coefficients.

(- 2x^2 + 2x^2) + (x + 4x) + (3 - 10)
0x^2 + 5x + (- 7)

3
Simplify the Result
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Finally, the obtained result of addition can be simplified. For example, since the coefficient before x^2-term is 0, this term is equal to 0 and can be omitted.

0x^2 + 5x + (- 7)
5x + (- 7)
5x - 7

Therefore, the sum is 5x-7. Note that the given polynomials are of degree 2. However, the new polynomial is of degree 1 because the x^2-terms canceled each other out. If they had not canceled each other out, the new polynomial would have also been of degree 2.

When adding or subtracting two polynomials of degrees m and n, the degree of the resulting polynomial is at most the equal to the degree of higher degree polynomial.

Extra

Vertical Format of Addition and Subtraction
Previously, polynomials were added by using the horizontal format. The vertical format of polynomial addition or subtraction will now be presented. Note that for this method, the like terms should be aligned vertically. &-2x^2 + x + 3 +& 2x^2 + 4x - 10 & 0x^2 + 5x - 7 By comparing the results, it can be concluded that no matter which format of addition is used, the sums are the same.

Exercises
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