Today, we learned about some operations with polynomials. Specifically, we learned how to do:

\(polynomial + other \hspace{0.15 cm} polynomial\),

\(constant \cdot polynomial\),

\(monomial \cdot polynomial\),

as well as some vocabulary.

There isn’t much intuition to go through today, as it’s mostly just distributing. I’ll still go through all of the stuff, just in case anybody is using this to learn.

Also, I'll teach you how to not panic when faced with a gigantic math thing.

Vocabulary

This vocabulary is just fun to me somehow. It’s probably because of the prefixes, which also make it easy to remember.

# terms name prefix
1 monomial mono-one
2 binomial bi-two
3 trinomial tri-three
4+ polynomial poly-many

There’s also the definition of degree: The highest power of a variable in a polynomial.

degree name prefix/root why
0 constant - - - it always returns a constant value
1 linear line it’s a line
2 quadratic quad - 4 it mainly deals with squares, which have 4 sides
3 cubic cube cubing something means to the power of 3
4 quartic ? probably just because “quad” was already used and “quart” sounds similar
5 quintic quint - 5 it’s degree 5
6 sextic/hexic hex - 6 it’s degree 6
7 septic/heptic hept/sept - 7 it’s degree 7
8 octic oct - 8 it’s degree 8
\(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\)

Other Things We Learned

\(polynomial + other \hspace{0.15 cm} polynomial\):

If you have this, just combine like terms.

\(constant \cdot polynomial\):

If you have this, distribute the constant to every term of the polynomial

\(monomial \cdot polynomial\):

If you have this, do the same thing as above. Just distribute it and use \(x^a \cdot x^b = x^{a+b}\) if needed.

How to Not Panic

If you ever see something like this:

Simplify: \(5[x+π-25(2^{18-2^{4}}-3)]+100x^2-(10+1)(3x)^2\)

And you only know how the basics, (probably) don’t panic.

Longer expressions or weird constants just make the problem take longer. They don’t make the problem any harder conceptually.

Here’s a 3-step guide to help you solve these types of problems.

Step 1: Skim through the problem.

Don’t panic. If you don’t see any new operations, you can definitely do this.

Step 2: Prepare.

You should get a sheet of scratch paper or some other place to record steps.

Step 3: Work through the steps.

Remember your order of operations!

Make sure not to skip any steps mentally, so that if the answer is wrong or doesn’t make sense, you can look back and find the error easily. From there, you can (usually) keep a lot of the other work that you did, only chaging the things necessary.

Let’s apply this guide.

Step 1: Skim through the problem

Result: Looks fine.

Step 2: Prepare.

Result: It’s right here.

Step 3: Work through the steps.

Result:

\(\hspace{0.55 cm} 5[x+π-25(2^{18-2^{4}}-3)]+100x^2-(10+1)(3x)^2\)\(\)

\(=5[x+π-25(2^{18-16}-3)]+100x^2-(11)(9x^2)\)\(\)

\(=5[x+π-25(2^{2}-3)]+100x^2-99x^2\)\(\)

\(=5[x+π-25(4-3)]+x^2\)\(\)

\(=5[x+π-25]+x^2\)\(\)

\(=5x+5π-125+x^2\)\(\)

\(=x^2+5x+5π-125\)\(\)

And that’s the answer.