Addition and Subtraction of Polynomials

Basic Concepts

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.

What Are Like Terms?

Like terms are terms that have the exact same variables and variable powers. The coefficients of these terms can be different.

Examples of Like Terms:

• $3x, \space -7x, \space -\frac{1}{5}x$ (all have the variable $x$ to the power of $1$)
• $\frac{1}{2}x^3, \space 4x^3, \space -2x^3$ (all have the variable $x$ to the power of $3$)
• $2x^2yz^3, \space \frac{3}{4}x^2yz^3, \space -5z^3x^2y$ (all have the variable $x$ to the power of $2$, $y$ to the power of $1$, and $z$ to the power of $3$)

Examples of Unlike Terms:

• $11x$ and $4x^2$ (different powers of $x$ )
• $4x^3$ and $-2x^4$ (different powers of $x$ )
• $4x^2$ and $-x^5$ (different powers of $x$ )
• $\frac{3}{4}x^2yz^3$ and $2x^2y$ (missing the variable $z^3$)
• $\frac{3}{4}x^2yz^3$ and $-7xy^2z^3$ (different powers of $x$ and $y$)

Polynomial Addition

To add two polynomials, we simply add the coefficients of like terms.

Horizontal Method:

1. Write both polynomials in parentheses connected by a plus sign.
2. Remove the parentheses.
3. Group the like terms.
4. Add the coefficients of each group of like terms (use the distributive property).

Example:

Find the result of $(2x^3 + 7x^2 + 3x + 5) + (6x^3 + 2x^2 + 4x + 1)$.

Polynomial Subtraction

To subtract two polynomials, we change the sign of each term in the polynomial being subtracted, then add them as usual.

Horizontal Method:

1. Write both polynomials in parentheses connected by a minus sign.
2. Remove the parentheses. Remember: change the sign of each term in the second parenthesis (distribute the negative sign).
3. Group the like terms.
4. Add the coefficients of each group of like terms.

Example:

Find the result of $(9x^3 + 4x^2 + 6x + 5) - (2x^3 + 3x^2 + 3x + 4)$.

Note: When subtracting, the negative sign in front of the parenthesis changes the sign of every term inside that parenthesis.

Vertical Method

Besides the horizontal method, polynomial addition and subtraction can also be done using the vertical method, similar to adding and subtracting regular numbers.

Steps:

1. Arrange both polynomials vertically.
2. Ensure like terms are aligned in the same column.
3. If a term is missing in one of the polynomials, leave a blank space or write a coefficient of $0$.
4. Add or subtract the coefficients in each column.

Example of Vertical Addition:

Example of Vertical Subtraction:

Both methods (horizontal and vertical) will yield the same answer. Choose the method that you find most comfortable and easiest to understand.

Graphical Addition and Subtraction of Polynomial Functions

Besides performing operations algebraically, we can also understand polynomial addition and subtraction visually through their graphs.

Suppose we have three graphs of polynomial functions:

Sketching the Graph of the Sum and Difference

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:

1. Choose several identical $x$ values on both graphs.
2. For each $x$ value, read the $y$ value from each graph. Let $y_f = f(x)$ and $y_g = g(x)$.
3. For addition ($f(x) + g(x)$): Calculate the value $y_{\text{new}} = y_f + y_g$.
4. For subtraction ($f(x) - g(x)$): Calculate the value $y_{\text{new}} = y_f - y_g$.
5. Plot the point $(x, y_{\text{new}})$.
6. Repeat for several other $x$ values.
7. 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 ($x$).

Example Sketches

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.

In the same way, you can sketch the graphs of $f(x) + h(x), \space g(x) + h(x), \space f(x) - h(x), \space \text{and} \space g(x) - h(x)$. The key is to add or subtract the heights (y-values) of the original graphs at each corresponding x-value.