Adding and Subtracting Polynomials

Adding and subtracting polynomials is easier than you might think - as long as you follow a simple method. Start by placing each polynomial in parentheses, using a + sign for addition or a − sign for subtraction. Then, remove the parentheses and combine like terms. Here's a quick example: $$ (a+b)+(b+c) = a+b+b+c = a+2b+c $$

Note. If you’re removing parentheses after a minus sign, don’t forget: you must change the sign of every term inside. For example: $$ (a+b)-(b+c) = a+b-b-c = a-c $$

How to Add Two Polynomials

Adding polynomials is a two-step process: first, write them in parentheses separated by a + sign. Then, drop the parentheses and combine like terms.

Example

Let’s add these two polynomials:

\[ (2a + 3b - c) + (-a + 2b - 3c) \]

First, remove the parentheses:

\[ 2a + 3b - c -a + 2b - 3c \]

Now, group the like terms together:

\[ (2a - a) + (3b + 2b) + (-c - 3c) \]

And simplify:

\[ a + 5b - 4c \]

The result is a trinomial!

How to Subtract Two Polynomials

Subtracting polynomials is just as straightforward. Write the polynomials in parentheses with a minus sign between them. Then, remove the parentheses - but this time, reverse the sign of every term in the second polynomial. Finally, combine like terms.

Example

Let’s subtract these polynomials:

\[ (2a + 3b - c) - (-a + 2b - 3c) \]

First, remove the parentheses and flip the signs in the second group:

\[ 2a + 3b - c + a - 2b + 3c \]

Group the like terms:

\[ (2a + a) + (3b - 2b) + (-c + 3c) \]

And simplify:

\[ 3a + b + 2c \]

There’s your result!

Key Takeaways

Here’s what you should remember about adding and subtracting polynomials:

• These operations are closed within polynomials
  When you add or subtract two polynomials, the result is always another polynomial. In other words, polynomials are closed under addition and subtraction.
• Subtraction is just a special case of addition
  Subtracting a polynomial is the same as adding its opposite. This is why both operations are often grouped under the concept of algebraic addition.

  Example. Take the difference: \[ (x - 4) - (x - 2) = x-4-x+2 = -2 \] Notice that the opposite of the second polynomial is \( -x + 2 \), so you could also write: \[ (x - 4) + (-x + 2) = x-4-x+2 = -2 \] - same result!

• Adding two opposite polynomials gives you zero
  A polynomial plus its opposite always equals zero.

  Example: \[ (x - 2) + (-x + 2) = 0 \]

• Subtracting a polynomial from itself also gives you zero
  Subtract any polynomial from itself, and you’ll always get zero.

  Example: \[ (x^2 + 3x) - (x^2 + 3x) = 0 \]

And that's the basics! Once you get the hang of it, adding and subtracting polynomials feels like second nature.