# How do you add two polynomials?

Feb 1, 2015

A polynomial is a sum of some powers of a certain variable, with some coefficient to multiply each power. Summing two polynomials simply means to sum the coefficients of the same powers, if this situations occour.

Let's say that your first polynomial is $1 + x + 2 {x}^{2} - 3 {x}^{3} + 15 {x}^{4}$, and the second is $- 3 {x}^{2} + 2 {x}^{3} + 5 {x}^{4} - 8 {x}^{5}$. If we add them, the result is
$1 + x + 2 {x}^{2} - 3 {x}^{3} + 15 {x}^{4} - 3 {x}^{2} + 2 {x}^{3} + 5 {x}^{4} - 8 {x}^{5}$. We can rearrange the terms so that the powers will be in order:

$1 + x + 2 {x}^{2} - 3 {x}^{2} - 3 {x}^{3} + 2 {x}^{3} + 15 {x}^{4} + 5 {x}^{4} - 8 {x}^{5}$
At this point, you simply need to notice that:

1. The constant term (i.e. $1$) appears only in the first polynomial, so we have nothing to sum
2. The same goes with the linear factor (i.e. $x$)
3. The quadratic factor (i.e. ${x}^{2}$) appears in both polynomial: in the first we have $2 {x}^{2}$, in the second we have $- 3 {x}^{2}$. Summing the two coefficient, we have $2 - 3 = - 1$. The result is $- {x}^{2}$
4. The cubic factor (i.e. ${x}^{3}$) appears in both polynomial: in the first we have $- 3 {x}^{3}$, in the second we have $2 {x}^{3}$. Summing the two coefficient, we have $- 3 + 2 = - 1$. The result is $- {x}^{3}$
5. The quartic factor (i.e. ${x}^{4}$) appears in both polynomial: in the first we have $15 {x}^{4}$, in the second we have $5 {x}^{4}$. Summing the two coefficient, we have $15 + 5 = 20$. The result is $20 {x}^{4}$
6. The quintic factor (i.e. ${x}^{5}$) appears only in the second polynomial, so we have nothing to sum

Finally, the answer is that the sum of the two polynomials is
$1 + x - {x}^{2} - {x}^{3} + 20 {x}^{4} - 8 {x}^{5}$