9.2: Addition and Subtraction of Polynomials
Suppose that two cars are having a race. The distance traveled by one car after \begin{align*}t\end{align*}
Watch This
Multimedia Link: For more explanation of polynomials, visit http://www.purplemath.com/modules/polydefs.htm  Purplemath’s website.
Guidance
To add or subtract polynomials, you have to group the like terms together and combine them to simplify.
Example A
Add and simplify \begin{align*}3x^24x+7\end{align*}
Solution: Add \begin{align*}3x^24x+7\end{align*}
\begin{align*}(3x^24x+7)+(2x^34x^26x+5)&=2x^3+(3x^24x^2 )+(4x6x)+(7+5)\\
&=2x^3x^210x+12\end{align*}
Example B
Subtract \begin{align*}5b^22a^2\end{align*}
Solution:
\begin{align*}(4a^28ab9b^2)(5b^22a^2)&=[(4a^2 (2a^2)]+(9b^25b^2)8ab\\
& = 6a^214b^28ab\end{align*}
Solving RealWorld Problems Using Addition or Subtraction of Polynomials
Polynomials are useful for finding the areas of geometric objects. In the following examples, you will see this usefulness in action.
Example C
Write a polynomial that represents the area of each figure shown.
(a)
(b)
Solution: The blue square has the following area: \begin{align*}y \cdot y=y^2\end{align*}
The yellow square has the following area: \begin{align*}x \cdot x = x^2\end{align*}
The pink rectangles each have the following area: \begin{align*}x \cdot y =xy\end{align*}
\begin{align*}\text{Test area} & = y^2+x^2+xy+xy\\
& = y^2 + x^2 + 2xy\end{align*}
To find the area of the green region we find the area of the big square and subtract the area of the little square.
The big square has an area of \begin{align*}y \cdot y =y^2\end{align*}
The little square has an area of \begin{align*}x \cdot x = x^2\end{align*}
Area of the green region \begin{align*}= y^2x^2\end{align*}
Guided Practice
Subtract \begin{align*} 4t^2+7t^33t5\end{align*}
Solution:
When subtracting polynomials, we have to remember to subtract each term. If the term is already negative, subtracting a negative term is the same thing as adding:
\begin{align*}6t+35t^3+9t^2(4t^2+7t^33t5)&= \\
6t+35t^3+9t^2(4t^2)(7t^3)(3t)(5)&=\\
6t+35t^3+9t^24t^27t^3+3t+5&=\\
(6t+3t)+(3+5)+(5t^37t^3)+(9t^24t^2)&=\\
9t+812t^3+5t^2&=\\
12t^3+5t^2+9t+8\\
\end{align*}
The final answer is in standard form.
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Addition and Subtraction of Polynomials (15:59)
Add and simplify.

\begin{align*}(x+8)+(3x5)\end{align*}
(x+8)+(−3x−5) 
\begin{align*}(8r^46r^23r+9)+(3r^3+5r^2+12r9)\end{align*}
(8r4−6r2−3r+9)+(3r3+5r2+12r−9) 
\begin{align*}(2x^2+4x12) + (7x+x^2)\end{align*}
(−2x2+4x−12)+(7x+x2) 
\begin{align*}(2a^2b2a+9)+(5a^2b4b+5)\end{align*}
(2a2b−2a+9)+(5a2b−4b+5) 
\begin{align*}(6.9a^22.3b^2+2ab)+(3.1a2.5b^2+b)\end{align*}
(6.9a2−2.3b2+2ab)+(3.1a−2.5b2+b)
Subtract and simplify.

\begin{align*}(t+15t^2)(5t^2+2t9)\end{align*}
(−t+15t2)−(5t2+2t−9) 
\begin{align*}(y^2+4y5)(5y^2+2y+7)\end{align*}
(−y2+4y−5)−(5y2+2y+7) 
\begin{align*}(h^7+2h^5+13h^3+4h^2h1)(3h^5+20h^33h^2+8h4)\end{align*}
(−h7+2h5+13h3+4h2−h−1)−(−3h5+20h3−3h2+8h−4) 
\begin{align*}(5m^2m)(3m^2+4m5)\end{align*}
(−5m2−m)−(3m2+4m−5) 
\begin{align*}(2a^2b3ab^2+5a^2b^2)(2a^2b^2+4a^2b5b^2)\end{align*}
(2a2b−3ab2+5a2b2)−(2a2b2+4a2b−5b2)
Find the area of the following figures.
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