A polynomial is a mathematical expression consisting of exponents, constants and variables.
Example:
The degree of the polynomial is the value of the highest exponent. For instance, in the above example, 3 is the highest exponent, so the degree of the polynomial is 3.
In a polynomial, the exponents need to be non-negative integers : 0, 1, 2, 3, … and we cannot divide by variables.
Examples that are not polynomials:
x ^{- 2} + x (we cannot have a negative exponent)
x/(x + 2) (We cannot have division by a variable)
x^{1/2 } (The exponent cannot be a fraction)
We can divide one polynomial by another polynomial using long division. Polynomial long division is similar to long division of numbers.
When we divide, the polynomials’ terms should be arranged in decreasing order of exponents, from the highest exponent to the lowest exponent.
For example, if we have x^{2} + x^{4} + 1, it should be rearranged as x^{4} + x^{2 }+ 1.
Suppose the question is (x^{4} + x^{2 }+ 1)/(x+2), then x^{4} + x^{2 }+ 1, the numerator, is called the dividend (the number that is going to be divided) and x + 2, the denominator, is called the divisor (the number that we are dividing by).
Please note that the following notations are equivalent:
There are 3 steps to doing polynomial long division:
and repeat the process.
Let us see an example!
In the above video, we didn’t have a remainder - remainder was 0. But that’s not always the case. The next video shows an example where we have a non zero remainder at the end.
Sometimes, we have missing terms that is we don’t have all the exponents in the dividend (numerator).
For example in the question, (x^{4} + x^{2 }+ 1) ÷ (x+2), in the dividend (numerator), we are missing terms with exponent 3 and exponent 1. The next video shows an example where we have missing terms in the dividend (numerator).
Now let's practice!