Like terms are really important when it comes to simplifying polynomial expressions.

But what are like terms?

In simple words, like terms are parts of an expression that share the same variable and have it raised to the same power.

For example, in the expression (3x^2 + 5x - 2x^2 + 7), the terms (3x^2) and (-2x^2) are like terms because they both have the variable (x) raised to the power of 2.

Why Combining Like Terms is Important

Combining like terms helps us make expressions simpler and easier to handle.

When we combine like terms, we add or subtract the numbers in front of those terms (called coefficients).

For example, in our earlier expression, we can combine (3x^2) and (-2x^2):

[ 3x^2 - 2x^2 = (3 - 2)x^2 = 1x^2 \text{ or just } x^2. ]

Steps to Simplify Polynomial Expressions

Here’s a simple way to simplify polynomials:

1. Find Like Terms: Look for terms that have the same variable raised to the same power.
2. Group the Terms: Put similar terms together so it's easier to see which can be combined.
3. Combine the Coefficients: Add or subtract the numbers in front of the like terms to simplify them.
4. Rewrite the Expression: After combining, write out the new and simpler expression.

Example

Let's look at a more complicated expression: (4x^3 - 2x^2 + 5 + 3x^3 + x^2 - 1).

1. Find Like Terms:

   • (4x^3) and (3x^3)
   • (-2x^2) and (x^2)
   • (5) and (-1)

2. Group the Terms:

   • (4x^3 + 3x^3)
   • (-2x^2 + x^2)
   • (5 - 1)

3. Combine the Coefficients:

   • (4 + 3 = 7 \Rightarrow 7x^3)
   • (-2 + 1 = -1 \Rightarrow -1x^2) or just (-x^2)
   • (5 - 1 = 4)

4. Rewrite the Expression:

So the simplified expression is:

[ 7x^3 - x^2 + 4. ]

By understanding like terms and how to combine them, you’ll get better at simplifying polynomial expressions. This is a really useful skill in algebra!