Addition / Subtraction of Polynomials

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It is straightforward to add or subtract two polynomials. We simply add or subtract terms of the same degree. For example:

\[\begin{array}{l} p\left( x \right):{x^2} + x + 1\\ q\left( x \right):{x^3} + 2{x^2} + \pi x + 2\\ \Rightarrow \;\;\;\;\;\;\;\;\;p\left( x \right) + q\left( x \right):{x^3} + 3{x^2} + \left( {1 + \pi } \right)x + 3\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;p\left( x \right) - q\left( x \right): - {x^3} - {x^2} + \left( {1 - \pi } \right)x - 1 \end{array}\]
 

Consider two polynomials of degrees m and n. Suppose that we add or subtract the two polynomials. Two cases now arise:

  1. If m and n are unequal, the resulting polynomial will be of a degree equal to the larger of m and n.

  2. If m and n are equal, the resulting polynomial will have a degree which is at the most equal to m and n, but it could also have a degree which is less than m and n.

Let’s understand these facts through the following examples:

(a) Suppose that

\[\begin{array}{l}p\left( x \right):{x^2} + x - 1\\q\left( x \right):{x^4} - {x^3} - 2\\ \Rightarrow \left\{ \begin{array}{l}{\rm{Deg}}\left( {p\left( x \right)} \right) = 2\\{\rm{Deg}}\left( {q\left( x \right)} \right) = 4\end{array} \right.\end{array}\]

Now,

\[\begin{array}{l}p\left( x \right) + q\left( x \right):{x^4} - {x^3} + {x^2} + x - 3\\p\left( x \right) - q\left( x \right): - {x^4} + {x^3} + {x^2} + x + 1\\ \Rightarrow \left\{ \begin{array}{l}{\rm{Deg}}\left( {p\left( x \right) + q\left( x \right)} \right) = 4\\{\rm{Deg}}\left( {p\left( x \right) - q\left( x \right)} \right) = 4\end{array} \right.\end{array}\]

(b) Suppose that

\[\begin{array}{l}p\left( x \right):{x^4} - {x^2} + 1\\q\left( x \right):{x^4} + {x^2} - 1\\ \Rightarrow \left\{ \begin{array}{l}{\rm{Deg}}\left( {p\left( x \right)} \right) = 4\\{\rm{Deg}}\left( {q\left( x \right)} \right) = 4\end{array} \right.\end{array}\]

Now,

\[\begin{array}{l}p\left( x \right) + q\left( x \right):2{x^4}\\p\left( x \right) - q\left( x \right): - 2{x^2} + 2\\ \Rightarrow  \left\{ \begin{array}{l}{\rm{Deg}}\left( {p\left( x \right) + q\left( x \right)} \right) = 4\\{\rm{Deg}}\left( {p\left( x \right) - q\left( x \right)} \right) = 2\end{array} \right.\end{array}\]

  \[\begin{array}{l}p\left( x \right) + q\left( x \right):2{x^4}\\p\left( x \right) - q\left( x \right): - 2{x^2} + 2\\ \Rightarrow \;\;\;\;\;\;\;\left\{ \begin{array}{l}{\rm{Deg}}\left( {p\left( x \right) + q\left( x \right)} \right) = 4\\{\rm{Deg}}\left( {p\left( x \right) - q\left( x \right)} \right) = 2\end{array} \right.\end{array}\]

Thus, in the case that the two polynomials have equal degrees, we see that there is a possibility of the highest degree terms cancelling each other out upon addition or subtraction, and so the resulting polynomial might have a degree less than the original degrees of the two polynomials.

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