Adding and Subtracting Polynomials

Adding and subtracting polynomials follows the same general principle of adding and subtracting numbers.

Though with polynomials we can only directly add and subtract ‘like terms’.

So we can do the following.     $x + x = 2x$     ,     $3a + 4a = 7a$     ,     $3x \space {\text{--}} \space x = 2x$

But we couldn’t add or subtract.     $x \space {\text{--}} \space y$     ,     $3a + 4b$

This really is the main thing to remember when you are dealing with adding and subtracting polynomials.



Examples    


(1.1) 

a)    $5x + x = 6x$         b)    $3a + 2a \space {\text{--}} \space a = 4a$         (c)    $2b + b \space {\text{--}} \space 2 = 3b \space {\text{--}} \space 2$



(1.2) 

a)    $x + 2y \space {\text{--}} \space x = 2y$         b)    $3a + 2a \space {\text{--}} \space a = 4a$         c)    $a + 8b + b \space {\text{--}} \space 3 = a + 9b \space {\text{--}} \space 3$



(1.3) 

$(\space 2x^{\tt{2}} + 3x \space {\text{--}} \space 4 \space) + (\space x^{\tt{2}} \space {\text{--}} \space 2x + 1 \space)$

Solution   

When larger polynomials are involved, it helps to group the like terms together before completing the addition or subtraction.

$(\space 2x^{\tt{2}} + 3x \space {\text{--}} \space 4 \space) + (\space x^{\tt{2}} \space {\text{--}} \space 2x + 1 \space)$

=   $2x^{\tt{2}} + x^{\tt{2}} + 3x \space {\text{--}} \space 2x \space {\text{--}} \space 4 + 1 \space$    =    $3x^{\tt{2}} + x \space {\text{--}} \space 3$



(1.4) 

$(\space x^{\tt{2}} + 6y \space) + (\space 5x^{\tt{2}} \space {\text{--}} \space y + 9 \space)$

Solution   

$(\space x^{\tt{2}} + 6y \space) + (\space 5x^{\tt{2}} \space {\text{--}} \space y + 9 \space)$

=   $x^{\tt{2}} \space {\text{--}} \space 5x^{\tt{2}} + 6y \space {\text{--}} \space ({\text{-}}y) \space {\text{--}} \space 9$    =    ${\text{-}}4x^{\tt{2}} + 7y \space {\text{--}} \space 9$

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Adding and Subtracting Polynomials,
Further Examples

Sometimes it can be required to add or subtract more than just two polynomials.

When this is the case, it’s often a better approach to use vertical addition or subtraction using appropriate rows and columns.



(2.1) 

$(\space x^{\tt{2}} + x + 2) + (\space 2x + 4 \space) + (\space x^{\tt{2}} + 3x \space {\text{--}} \space 2 \space)$

Solution   

We line up the polynomials to be added in appropriate columns,
so each like term is in line with its other like terms.

$\begin{array}{r} \space x^{\tt{2}} + \space\space x + 2\\ 5x^{\tt{2}} \space\space {\text{--}} \space 3x + 9\\ {\color{green}{\bf{+}}} \space \space \space \space \space 2x^{\tt{2}} + 4x \space\space {\text{--}} \space 6\\ \hline \\ \end{array}$

Then carry out the relevant addition or subtraction and write the result below, where we should end up with the correct answer.

$\begin{array}{r} \space x^{\tt{2}} + \space\space x + 2\\ 5x^{\tt{2}} \space\space {\text{--}} \space 3x + 9\\ {\color{green}{\bf{+}}} \space \space \space \space \space 2x^{\tt{2}} + 4x \space\space {\text{--}} \space 6\\ \hline 8x^{\tt{\tiny{2}}} + 2x + 5 \end{array}$

$(\space x^{\tt{2}} + x + 2) + (\space 2x + 4 \space) + (\space x^{\tt{2}} + 3x \space {\text{--}} \space 2 \space) \space\space = \space\space 8x^{\tt{2}} + 2x + 5</font>$




(2.2) 

$(\space 2x^{\tt{2}} + x \space {\text{--}} \space 3) + (\space 2x + 4 \space) + (\space x^{\tt{2}} + 3x \space {\text{--}} \space 2 \space)$

Solution   

Here there is no $x^{\tt{2}}$ like term in the second polynomial to be added.
When this happens, we either leave the entry blank in the vertical addition column, or one can write zero.

$\begin{array}{r} \space \space 2x^{\tt{2}} + \space\space x \space\space {\text{--}} \space 3\\ 2x + 4\\ {\color{blue}{\bf{+}}} \space \space \space \space \space x^{\tt{2}} + 3x \space\space {\text{--}} \space 2\\ \hline 3x^{\tt{\tiny{2}}} + 6x \space\space {\text{--}} \space 1 \end{array}$

$(\space 2x^{\tt{2}} + x \space {\text{--}} \space 3) + (\space 2x + 4 \space) + (\space x^{\tt{2}} + 3x \space {\text{--}} \space 2 \space) \space\space = \space\space 3x^{\tt{2}} + 6x \space\space {\text{--}} \space 1$




(2.3) 

$(\space x^{\tt{2}} + 3x \space {\text{--}} \space 1) \space {\text{--}} \space (\space 5x^{\tt{2}} \space {\text{--}} \space 2x + 6 \space) + (\space 2x^{\tt{2}} + 8x \space {\text{--}} \space 7 \space)$

Solution   

With this example we subtract the terms in the middle row of the sum.

$\begin{array}{r} \space \space x^{\tt{2}} + 3x \space\space {\text{--}} \space\space 1\\ {\color{darkred}{\bf{{\text{--}}}}} \space \space \space \space \space 5x^{\tt{2}} + 2x \space + 6\\ {\color{blue}{\bf{+}}} \space \space \space \space \space 2x^{\tt{2}} + 8x \space\space {\text{--}} \space\space 7\\ \hline {\text{-}}2x^{\tt{\tiny{2}}} + 9x \space\space {\text{--}} 14 \end{array}$

$(\space x^{\tt{2}} + 3x \space {\text{--}} \space 1) \space {\text{--}} \space (\space 5x^{\tt{2}} \space {\text{--}} \space 2x + 6 \space) + (\space 2x^{\tt{2}} + 8x \space {\text{--}} \space 7 \space) \space\space = \space\space {\text{-}}2x^{\tt{2}} + 9x \space {\text{--}} \space 14$

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