Have you ever been caught in a rainstorm without a coat and thought,
“I wish I could’ve seen this coming.”
Or maybe you’ve experienced an even bigger betrayal
-- you checked your weather app and it told you there’s only a 10% chance of rain, so
you risked it, and now you’re drenched.
Meteorologists use math to try and describe the world and predict the future
-- same as biologists, economists, political pollsters, professional gamblers, or almost
anything else you might dream of being.
Mathematicians call this modeling.
But any model that’s trying to capture a world filled with chaotic particles
-- not to mention humans who make irrational decisions and live complicated lives -- will
look pretty messy.
Instead of turning to a deck of tarot cards or a crystal ball to make things simpler,
we can use polynomials.
A polynomial is an algebraic quantity where the variables involved have only whole, non-negative
exponents,
like in x^2 and x^5. And they can actually approximate a lot of the messy models we might
create to get us some answers.
So today we’ll learn the basics of manipulating polynomials. Let’s see what mathematicians
use to tell the future.
G’Day, I’m James Tanton, and this is Study Hall Algebra presented by Arizona State University
and Crash Course.
Before we dive in, let’s remember the difference between an algebraic expression
and an algebraic equation.
Algebraic expressions are generic quantities that combine numbers and variables.
5x+3 is a polynomial expression because it is just a quantity, and it’s a polynomial
because the variables have only whole, non-negative exponents.
And here, x can be anything.
But algebraic equations are trying to say something specific
-- that there is equality between two quantities.
For example, 5x+3 = 13 is a polynomial equation with a single solution x=2.
Now we won’t pretend you’ll ever multiply or divide complicated expressions by hand
after algebra class.
But you will need to know how to tell a computer what to do and be able to
interpret whatever it spits out.
Adding and subtracting polynomials is basically just collecting like terms,
or terms with the same variable raised to the same exponent.
It’s fiddly work, but straightforward if you have a system.
A lot of the work comes down to paying attention to the coefficients,
or the numbers multiplying our variables. Like the 4 in 4x.
But multiplying polynomials is more convoluted, especially for long expressions.
The box method -- which we first saw with negative numbers
-- is excellent for keeping careful track of how each term gets distributed, like for
multiplying great big galumphing polynomials.
If we have to multiply something like 3x^2 + 2x – 7 by 4x^2 + 3x + 5,
well – I don’t know about you, but if I tried to do that in my head, I’d fall down.
But if we divide this up into boxes, nothing gets missed. 2
For the first row, 3x^2 times 4x^2 is 12x^4, 2x times 4x^2 is 8x^3, –7 times 4x^2 is –28x^2.
Order doesn’t matter as long as every box gets filled in.
To finish, we can clean up by adding all the terms together, reordering them from highest
power to lowest power, and collecting like terms.
Multiplication is all about organization. But polynomial division is all about solving
mysteries.
We’re actually working with two types of mysteries here.
In one, we’re given a polynomial and directly instructed to divide it by another.
And in the other we’re factoring, or figuring out what polynomials multiply together to
produce our original polynomial.
Somewhere in the Math Goggles, there’s a magnifying glass setting that will help us
make these deductions.
The product: 6x^3 + 6x^2 + 3x – 15. The criminal mastermind: 2x^2 + 4x + 5.
We just need our accomplice to complete the puzzle… and we’re on the case.
We’ll start by lining everything up just as it was at the scene.
3 Unfortunately, some of our evidence is compromised.
We know we’ve added or subtracted some terms together to get that 3x, for example, so let’s
not fill it in just yet.
But our first term and our last term are almost always safe to fill in because they don’t
come from combining like terms.
Examining their coefficients might help us bust this case wide open.
What could turn 2x^2 into 6x^3?
Well, the coefficient must have been 3, since 2 times 3 is six.
And to turn x^2 into x^3 there must’ve been an x involved too.
So odds are this first guy works under the name of 3x.
This term had its mitts all over the first row.
3x*4x is 12x^2. 3x*5 is 15x. As soon as we pin down one of these characters,
we can round them all up.
Let’s put the screws to that –15.
What could have come along and turned that innocent 5 into a –15? A –3 must have done it.
And when we pencil that in, suddenly we’ve got everybody. –3*2x^2 is –6x^2, and
–3*4x is -12x.
Now it’s time to check that all these stories add up.
6x^3: good. 12x^2 – 6x^2: gives us 6x^2. Checks out so far. 15x – 12x gives us 3x,
and we can close the case.
Factoring is also like working backwards from the scene of the crime, but this time we only
have one polynomial to start with.
Let’s say we have something like x^2 + 2x – 84, and we want to split this up into
two smaller polynomials.
Two quantities were multiplied together here to produce an x^2, and our first guess is
that it comes from x times x.
What about that –8? Guessing is harder here: it could be the product of –1 times 8, or
1 times –8. Or –2 and 4. Or 2 and –4.
To decide, we can interview the coefficient of the 2x term.
It suggests that the constants aren’t as far apart as 1 and 8, because –8x + 1x add
to -7x, or 8x – 1x is 7x.
So we want a pair with a small gap, like 2 and 4.
The larger number is probably positive because 2 is positive and a large number minus a small
number will still be positive.
So we’ll guess that our factors are (x + 4) and (x – 2).
We might have to try a couple of options, but that’s absolutely fine.
Every good mystery has a few red herrings.
As long as we establish a list of suspected factors and we’re patient,
we’ll get them in the end.
Factoring might seem like complicated rewriting, but it reveals important secrets about whatever
pattern we’re trying to describe that the expanded polynomial might be hiding.
x^2 + 2x - 8 wasn’t giving up that x = -4 and x=2 actually make the polynomial equal 0 --
but the factored version was ready to spill the beans.
So after all the work we’ve done expanding number systems and exploring different rules,
can we solve every type of equation out there?
Figuring out when we’re “done” with numbers is definitely -- as mathematicians
would say -- non-trivial, or not obvious.
We know there must be something else out there when we come across something like x^2 = –9.
Nothing that we have solves that equation.
So we need to use a number that doesn’t actually exist in the real world.
Mathematicians filled this gap by introducing the number i, for imaginary.
I know, it doesn’t look like a number, but we had to come up with something with the
fanciful property that if you square it, you get -1.
Sometimes we get backed into a corner by the simultaneous need to take the square root
of a negative number
… and the fact that there is no such thing as the square root of a negative number.
But i makes it all possible.
Take x^2 = –9. If this were x^2 = positive 9, we could say that there are two solutions,
x = 3 and x = –3.
But now we want something squared to be -9. Stick an i in, and instead try x = 3i
and x = -3i.
(3i) squared is (3 times i) times (3 times i), which is 9 times i^2, which is 9 times
–1, which is -9. Woo!
And (-3i)^2 is (-3 times i) times (-3 times i), which is 9 times i^2 again, which is -9!
Double woo!
One of the greatest surprises in the human story of mathematics is that imaginary numbers
are enough to solve all polynomial equations.
We’ve needed more and more new ideas and techniques
– negative numbers, rational expressions, irrational numbers, logarithms – as we come
up against different types of equations.
So we might wonder if even more new, weird, fantastical numbers are needed to solve even
more equations.
But with imaginary numbers, mathematicians were able to prove the Fundamental Theorem
of Algebra, which guarantees that every polynomial can be solved with an integer, a fraction,
an irrational, or an imaginary number.
Knowing that all equations have solutions means we can go ahead and create algorithms,
or write programs, for finding solutions to equations without fear of the program crashing.
While these imaginary solutions might not have a real world meaning, they do make sure
all the math works.
So now we’re polynomial masters, right?
We can solve, multiply, and factor almost anything algebra throws at us.
But, it’s always possible to get stuck. Here are some tips if you do.
When adding and subtracting polynomials, it’s important to stay organized.
Stack your polynomials and align variables with the same power.
Rectangles are still our friends! The box method works for both multiplication and division.
Filling in the first and last terms of the dividend will help you to sleuth out what
your factors are.
If you need to take the square root of a negative number, there’s an imaginary number lurking in there.
Remember that you will still get both a positive and a negative version of the root.
Next time we’ll go deeper into polynomials and look at one of the most famous – and
infamous – types: the quadratic equation.
Until then, cheers!
Thanks for watching Study Hall Algebra, which is produced by Arizona State University and
the Crash Course team at Complexly.
If you liked this video and want to keep learning with us here in Study Hall, be sure to subscribe.
You can learn more about ASU and the videos produced by Crash Course
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See you next time!