How do I solve polynomials

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

Also they can have one or more terms, but not an infinite number of terms.

These are polynomials:

3x

x − 2

−6y2 − ( 79)x

3xyz + 3xy2z − 0.1xz − 200y + 0.5

512v5 + 99w5

5

(Yes, "5" is a polynomial, one term is allowed, and it can be just a constant!)

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:

monomial, binomial, trinomial

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.

Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines.

Example: x4−2x2+x

x^4-2x^2+x  

See how nice and

smooth the curve is?

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:

4x3-x-3The 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

You don't have to use Standard Form, but it helps.