It’s Professor Dave, let’s work with polynomials.
So we’ve defined the polynomial as a concept, it’s an expression or equation involving
multiple terms that each include a variable raised to some positive whole number exponent,
which may also be multiplied by some coefficient.
These are listed in descending order, ending with any constant, which can be thought of
as the X to the zero term, since X to the zero equals one.
Now that we know what these are, we have to be able to manipulate them and perform various
operations.
Let’s start with the simplest operations, addition and subtraction.
When we add and subtract polynomials, the key is to combine like terms, because we want
a single X to the fourth term, a single X cubed term, and so on, for any exponents that
happen to be present in the expressions.
When we combine like terms, this is the same application of the distributive property that
we have already learned.
Just as five X plus four X is equal to the quantity of five plus four times X, or nine
X, this will apply to five X cubed plus four X cubed.
We can ignore the exponent for a moment, because we can simply recognize that they match.
If they match, it doesn’t matter what the term is, we will only manipulate the coefficient.
Five X cubed plus four X cubed may as well be five apples plus four apples, where the
symbol of the apple can be taken to represent X cubed.
Once we see things that way, it is easy to see how these terms add together to become
nine X cubed.
This is also apparent when we apply the distributive property, as X cubed could operate across
the sum of five plus four.
Now let’s add two whole expressions together.
Four X cubed plus two X squared minus six X plus nine will be one expression, and the
other is X to the fourth minus three X squared plus six X minus four.
If we want to add these together, there are several ways we could do it, and whichever
way works best for you is just fine.
We could just write a new expression with all of these terms in a line if we wanted.
Then it’s just a matter of identifying like terms and adding them together.
However, a slightly tidier approach, if you value that sort of thing, could involve putting
one expression over the other, so that we can do arithmetic just the way we did with
small numbers.
If we do this, it is important to notice the degree on each term, and make sure the degrees
are lined up properly, so that we add or subtract the correct terms.
This could involve leaving a blank space if a particular degree is not present in an expression,
like this one, which does not have an X cubed term.
This kind of bookkeeping will help us avoid making a careless error.
Now with everything lined up, we just add.
X to the fourth is by itself, so we leave it alone.
Same with four X cubed, that will stay as is.
Then we have two X squared minus three X squared, which leaves us with negative X squared.
Negative six X plus six X leaves us with zero for the X term, and nine minus four is five.
So X to the fourth plus four X cubed minus X squared plus five will be the answer.
Subtraction is essentially the same, we just have to be a little more careful about our
signs.
Let’s say we keep each expression as it is, but we subtract the second one from the
first.
Now we are subtracting X to the fourth from zero X to the fourth.
That gives negative X to the fourth.
Then it’s four X cubed minus zero, so that term stays as it is.
Then it’s two X squared minus negative three X squared.
Remember that subtracting a negative number is the same as adding it, so this becomes
two X squared plus three X squared, or five X squared.
Negative six X minus six X is negative twelve X, and nine minus negative four is thirteen.
And this is how we subtract one polynomial from another.
So that’s all there is to it for the addition and subtraction of polynomials.
Things get trickier when we try to multiply polynomials, but before we move on to that,
let’s check comprehension.