Collecting Polynomials

This is not difficult in principle, but it can become messy if you are not careful with your signs.

You can only add or subtract values with the same powers of x.

Be careful: this means that you CANNOT add 3x⁴ and 3x³ because they have different powers of x and are therefore completely different values.

You also CANNOT add 3x² and 3r² because they are different variables and so different values.

Here is a fairly basic example: add 3x³ + 5x to 2x³ -4x

It helps to collect the two expressions into brackets first:

(3x³ + 5x) + (2x³ -4x)

Now you can sort out the signs more easily:

3x³ + 5x + 2x³ -4x

As it happens there is no sort out needed here!

Now collect the like terms:

(3x³ + 2x³) + (5x - 4x)

= 5x³ + x

It does get a lot more fiddly than this sometimes:

(4x⁴ + 5x³ - 3x²) - (2x⁴ - 6x³ + x² + 9)

But use the same principles. Here the two expressions are already in brackets so we can SORT THE SIGNS OUT:

4x⁴ + 5x³ - 3x² - 2x⁴ + 6x³ - x² - 9

What we've done here is simply say "all the values in the second polynomial are being subtracted and we all know what to do with a - and a + or a - and another -". We've effectively subtracted the two expressions here by reversing the signs in the second expression.

You might want to look back at that again.

OK. Now we can collect the like terms.

4x⁴ - 2x⁴ + 5x³ + 6x³ - 3x² - x² - 9

= 2x⁴ + 11x³ - 4x² - 9