Math11: Chapter 7- More Algebra

Polynomials

A polynomial looks like this:

example of a polynomial, this one has

3 terms

Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms"

A polynomial can have:

constants (like 3, -20, or ½)

variables (like x and y)

exponents (like the 2 in y²), but

only 0, 1, 2, 3, ... etc are allowed

that can be combined using addition,

subtraction, multiplication and

division ...

... except ...

... not division by a variable (so

something like 2/x is right out)

So:

A polynomial can have constants, variables and exponents,

but never division by a variable.

Polynomial or Not?

These are polynomials:

x - 2

-6y² - (7/9)x

3xyz + 3xy²z - 0.1xz - 200y + 0.5

512v⁵+ 99w⁵

(Yes, even "5" is a polynomial, one

term is allowed, and it can even be

just a constant!)

And these are not polynomials

3xy^-2 is not, because the exponent is "-2" (exponents can only be 0,1,2,...)
2/(x+2) is not, because dividing by a variable is not allowed
1/x is not either
√x is not, because the exponent is "½" (see fractional exponents)

But these are allowed:

x/2 is allowed, because you can divide by a constant
also 3x/8 for the same reason
√2 is allowed, because it is a constant (= 1.4142...etc)

Monomial, Binomial, Trinomial

There are special names for

polynomials with 1, 2 or 3 terms:

How do you remember the names?

Think cycles!

There is also quadrinomial (4 terms)

and quintinomial (5 terms),

but those names are not often used.

Can Have Lots and Lots of Terms

Polynomials can have as many terms as

needed, but not an infinite number

of terms.

Variables

Polynomials can have no variable at all

Example: 21 is a polynomial. It has just one term, which is a constant.

Or one variable

Example: x⁴-2x²+x has three terms, but only one variable (x)

Or two or more variables

Example: xy⁴-5x²z has two terms, and three variables (x, y and z)

What is Special About Polynomials?

Because of the strict definition,

polynomials are easy to work with.

For example we know that:

If you add polynomials you get a polynomial
If you multiply polynomials you get a polynomial

So you can do lots of additions and

multiplications, and still have a

polynomial as the result.

You can also divide polynomials (but

the result may not be a polynomial).

Degree

The degree of a polynomial with only

one variable is the largest exponent

of that variable.

Example:

The Degree is 3 (the largest exponent of x)

For more complicated cases,

read Degree (of an Expression).

Standard Form

The Standard Form for writing a

polynomial is to put the terms with the

highest degree first.

Example:

Put this in Standard Form:
3x² - 7 + 4x³ + x⁶
The highest degree is 6, so that goes

first, then 3, 2 and then the constant

last:

x⁶ + 4x³ + 3x² - 7

Division of polynomials:

Long division for polynomials works in much the same way:

First, I set up the division:For the moment, I'll ignore the other terms and look just at the leading x of the divisor and the leadingx² of the dividend.
If I divide the leading x² inside by the leading x in front, what would I get? I'd get an x. So I'll put an xon top:
Now I'll take that x, and multiply it through the divisor, x + 1. First, I multiply the x (on top) by thex (on the "side"), and carry the x² underneath:
Then I'll multiply the x (on top) by the 1 (on the "side"), and carry the 1x underneath:
Then I'll draw the "equals" bar, so I can do the subtraction.To subtract the polynomials, I change all the signsin the second line...
...and then I add down. The first term (the x²) will cancel out:
I need to remember to carry down that last term, the "subtract ten", from the dividend:
Now I look at the x from the divisor and the new leading term, the –10x, in the bottom line of the division. If I divide the –10x by the x, I would end up with a –10, so I'll put that on top:
Now I'll multiply the –10 (on top) by the leading x(on the "side"), and carry the –10x to the bottom:
...and I'll multiply the –10 (on top) by the 1 (on the "side"), and carry the –10 to the bottom:
I draw the equals bar, and change the signs on all the terms in the bottom row:
Then I add down: