# Polynomials

### Topics

## Introduction to :

By now, your head might be spinning. Very reasonably, we might add. If so, focus on a single spot in the room and *don't look away*. Common factors, grouping, squares, sums, differences...gah! It's harder to keep all these straight than it is a 6-pound bowling ball.

When given a bunch of polynomials and asked to factor them, how do we know which method to use for each polynomial? Should we flip a coin? Draw straws? Magic 8-Ball?

To add one more coat of crazy paint, some polynomials can't be factored at all. If a polynomial can't be factored, it's called **prime**. A prime polynomial is like a prime number: there's no way to break down a prime number into a product of smaller numbers, and there's no way to break down a prime polynomial into a product of simpler polynomials. Because prime numbers and polynomials are easier to work with, this result is optimal. In other words, it's an optimal prime. Dude. Hopefully that bad pun didn't *transform* your opinion of us.

We have enough factoring methods now that if you try and try to factor a polynomial to no avail, the polynomial is probably prime.

Back to the question: Which method for factoring will be your BFF? Answer: Doesn't matter, as long as you have the right answer. If you want to be sure you're right, then check your answers using the distributive property and/or FOIL. This way, you can have multiple BFFs. Just don't tell the others.

If a problem tells you specifically what method to use, be a genius and use that method! Otherwise, start with the easiest method and work your way through to the hardest. Which method is "easiest" or "hardest" depends on personal preference. You may prefer trial-and-error, while your friend might like factoring by grouping. Although, it's a wonder you are friends, considering how little you seemingly have in common. If you can't connect on a common factoring method, what *can* you connect on?

Here's our suggested plan of attack when faced with a rogue polynomial:

- If there's a common factor for all the terms, factor it out.

- See if the polynomial is recognizable as a product of a sum and difference or as a square of a binomial.

- If the polynomial is written with 4 terms, try factoring by grouping.

- Try factoring by grouping or trial-and-error, whichever you like better, or whichever one is "calling to you."

We say a polynomial is *factored completely* when it can't be factored any more. Imagine that. For example, 2*x*^{2} + 4*x* factors as (2)(*x*^{2} + 2*x*), but this isn't a complete factoring. You wouldn't mow only half the lawn, would you? If your allowance depended on it, we mean.

We can factor further, so we pull out an *x* so (2*x*)(*x* + 2).

#### Exercise 1

Factor 9*x*^{2}y^{2} - 24*xy*^{2} + 16y^{2} completely or state that it is prime. If you are able to factor the polynomial, what technique(s) or method(s) did you use?

#### Exercise 2

Factor 25*x*^{8} - 49*x*^{2} completely or state that it is prime. If you are able to factor the polynomial, what technique(s) or method(s) did you use?

#### Exercise 3

Factor *x*^{2} + 4 completely or state that it is prime. If you are able to factor the polynomial, what technique(s) or method(s) did you use?

#### Exercise 4

Factor 7*x*^{2} - 20*x* - 3 completely or state that it is prime. If you are able to factor the polynomial, what technique(s) or method(s) did you use?

#### Exercise 5

Factor 5*x*^{2} - 35*x* completely or state that it is prime. If you are able to factor the polynomial, what technique(s) or method(s) did you use?