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Definition of polynomial and key terms. Operations with polynomials, and thinking about degree.
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A polynomial is a combination of positive integer powers xn:
This can also be expressed as a summation:
All the afactors are simply arbitrary constants. We call them the coefficients of the polynomial.
The degree of a polynomial is the highest power of x that appears, which is n for polynomials of the above form.
The degree of a product of polynomials is the sum of the degrees:
Each term akxk in a polynomial is associated with a specific power of x. We say that
and
When we have a product of polynomials, it is possible to find the coefficient of a specific term without expanding the entire product. To do this, we consider which terms from the first polynomial multiplies with which term from the second polynomial to give the desired power of x.
If two polynomials are equal for all x∈R (not just an intersection), then all their coefficients are equal: