A polynomial in the variable x is a mathematical expression involving the sum of terms. It is in the form of ax^n, where a is a coefficient to the power of n, which has to be a non-negative number. Division by variables are not allowed.
Polynomials are arranged systematically, in acceding or descending order according to the power of x.
The degree or the degree of freedom, is the highest power in the polynomial.
When the LHS of an equation is the same as the RHS of the equation for all values of x, it is called an identity. A "≡" symbol is used.
If both expression co-incide on a graph, they are an identity. (refer to mason's post below ^^)
Polynomials will be affected by the values of x, hence, x² -1 / x-1 is not an identity of x+1. This is because x in this equation cannot equal to 1 and the definition says it has to be true for all values of x.
Note: -1³ ≠ (-1)³ , remember to insert the brackets when necessary.
DIVISION OF POLYNOMIALS
Using the primary school analogy of long division, we can gather dividend ≡ divisor x quotient + remainder.
Sometimes there will be "missing terms" (example: there may be an x³, but no x²). In that case, include the missing terms with a coefficient of zero.
You also can use Synthetic Division. The divisor should be linear and the first step is to make it equal to zero to find the value of x and put it on the left side.
Lastly, you could just try to input factors to work out the remainder.