The operations of addition and subtraction on polynomials are essentially the same as for other algebraic forms: we can only add or subtract like terms.
Besides the horizontal method, polynomial addition and subtraction can also be done using the vertical method, similar to adding and subtracting regular numbers.
Steps:
Arrange both polynomials vertically.
Ensure like terms are aligned in the same column.
If a term is missing in one of the polynomials, leave a blank space or write a coefficient of 0.
Add or subtract the coefficients in each column.
Example of Vertical Addition:
+2x36x38x3+7x2+2x2+9x2+3x+4x+7x+5+1+6
Example of Vertical Subtraction:
−9x32x37x3+4x2+3x2+1x2+6x+3x+3x+5+4+1
Both methods (horizontal and vertical) will yield the same answer. Choose the method that you find most comfortable and easiest to understand.
Without needing the exact equations of functions f, g, or h, we can sketch the graph of their sum (e.g., f(x)+g(x)) or difference (e.g., f(x)−g(x)) as follows:
Choose several identical x values on both graphs.
For each x value, read the y value from each graph. Let yf=f(x) and yg=g(x).
For addition (f(x)+g(x)): Calculate the value ynew=yf+yg.
For subtraction (f(x)−g(x)): Calculate the value ynew=yf−yg.
Plot the point (x,ynew).
Repeat for several other x values.
Connect the new points with a smooth curve.
Why does this work?
Because the definition of function addition or subtraction is to add or subtract their output values (y) for each corresponding input value (x).
Here are example sketches of the graphs resulting from the sum f(x)+g(x) and difference f(x)−g(x), obtained by vertically adding/subtracting the y-values for each x.
Graph of f(x)+g(x), f(x)=x+1, and g(x)=0.5x4−2x2+1
The result of f(x)+g(x) is x+1+0.5x4−2x2+1
Graph of f(x)−g(x), f(x)=x+1, and g(x)=0.5x4−2x2+1
In the same way, you can sketch the graphs of f(x)+h(x),g(x)+h(x),f(x)−h(x),andg(x)−h(x). The key is to add or subtract the heights (y-values) of the original graphs at each corresponding x-value.
