# At a Glance - Combining Polynomials

## Adding Polynomials

To add polynomials, all we do is combine like terms in the same way as we would with any other expression. Sorry, we know you were probably hoping for something new and wildly different, but you'll need to settle for doing things the way you already know how.

### Sample Problem

What's the sum of the polynomials 6*x*^{3} + 5*x* + 7 and 8*x*^{2} – 9*x* – 1?

(6x^{3} + 5x + 7) + (8x^{2} – 9x – 1) | = | (6x^{3}) + (8x^{2}) + (5x – 9x) + (7 – 1) |

= | 6x^{3} + 8x^{2} – 4x + 6 |

Quick flashback: Remember how, back in the Jurassic period when we didn't have access to calculators, we used to stack up whole numbers in order to add them, like this?

We can do something similar with polynomials. We stack the polynomials on top of each other so that terms with the same degree line up vertically. This can help us keep track of all the terms. It can also help us maximize our closet space.

### Sample Problem

Find the polynomial sum:

(5*x*^{7} + 6*x*^{4} + 3*x*^{2} – 4*x* – 8) + (-3*x*^{7} + *x*^{5} – *x*^{4} + 8*x* + 11)

First, write the polynomials on top of each other so like terms line up:

Then, add down each column:

2*x*^{7} + *x*^{5} + 5*x*^{4} + 3*x*^{2} + 4*x* + 3

If you like the vertical stacking, go for it. If not, forget we mentioned it. When adding polynomials, use whatever method works for you. Just remember to combine like terms. Don't go crazy and add together everything you see regardless of what degree they are. That's plain irresponsible.

## Subtracting Polynomials

Finding the difference between two polynomials is like finding the difference between any other two expressions. We visited this idea in the section on Getting Rid of Parentheses. We should visit it again soon, because it's been calling a lot lately and giving us a huge guilt trip.

### Sample Problem

What is (6*x*^{3} + 5*x*^{2} – 8*x*) – (*x*^{2} + 3*x* – 12)?

This is equivalent to:

(6*x*^{3} + 5*x*^{2} – 8*x*) + (-1)(*x*^{2} + 3*x* – 12)

After distributing the (-1), the original subtraction simplifies to:

(6*x*^{3} + 5*x*^{2} – 8*x*) – *x*^{2} – 3*x* + 12

Notice that we now have a positive 12 at the end thanks to that whole "double negative" thing. We no longer need the first set of parentheses, so we can get rid of them...real quiet-like...

6*x*^{3} + 5*x*^{2} – 8*x* – *x*^{2} – 3*x* + 12

Now we put like terms next to each other to find:

6*x*^{3} + (5*x*^{2} – *x*^{2}) + (-8*x* – 3*x*) + 12

And finally combine like terms:

6*x*^{3} + 4*x*^{2} – 11*x* + 12

A common mistake in subtracting polynomials is forgetting to subtract every term and instead only subtracting the first term. Forgetting leads to things like milk in the pantry and ketchup on your salad, so here's a way to help: it's called the **"flip and add" method**. First, we *flip* the sign of every term in the second polynomial to find a new polynomial. Then we *add *that new polynomial to the first one. This helps us to remember that when we subtract a polynomial, we subtract every term, not only the first one. It's like shampoo and conditioner in one, except more useful on a math test.

As with addition, there are different ways to organize your work for polynomial subtraction. Use your favorite way and work carefully to avoid minus sign mistakes, also known as the most common mistake in algebra. Also try to avoid the second most common mistake in algebra: discussing mixed numbers in mixed company.

#### Exercise 1

Add the following polynomials:

(3*x*^{27} + 4*x*^{20} – 6*x*^{11} + *x*) + (*x*^{26} + 6*x*^{20} + 8*x*^{11} – *x*)

#### Exercise 2

Add the following polynomials:

(-2*x*^{10} + 13*x*^{7} – 4*x* + 9) + (7*x*^{10} – 12*x*^{7} – 8*x* – 1)

#### Exercise 3

Add the following polynomials:

(17*x*^{2} + 14*x* + 3) + (-3*x*^{2} + 10)

#### Exercise 4

Add the following polynomials:

(4*x* + *xy* – 2*y*) + (3*x* – 4*xy* + *y*^{2})

#### Exercise 5

Add the following polynomials:

(2*x*^{2}*y* + 4*xy*^{2}) + (3*xy*^{2} – *x*^{2}*y*)

#### Exercise 6

Subtract the following polynomials:

(4*x*^{2} – 3*x* + 9) – (2*x*^{2} + *x* + 1)

#### Exercise 7

Subtract the following polynomials:

(7*x*^{3} + 2*x* – 4) – (-*x*^{3} + 3*x* – 5)

#### Exercise 8

Subtract the following polynomials:

(-10*x*^{4} – 6*x*^{3} + *x*) – (-2*x*^{4} – *x*^{2} + 4*x*)

#### Exercise 9

Subtract the following polynomials:

(2*xy* + *x*^{2}*y* + 3*xy*^{4}) – (*xy* – *xy*^{2} + 2*xy*^{4})

#### Exercise 10

Subtract the following polynomials:

(-13*x*^{2}*y* + 2*xy* – *y*^{2}) – (-*x*^{2}*y* + 2*xy* + *y*^{2})